Theorem: mem_generatePiSystem_iUnion_elim
File: Mathlib/MeasureTheory/PiSystem.lean

tactic: induction' h_t with s h_s s t' h_gen_s h_gen_t' h_nonempty h_s h_t'
state_before: α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
h_t : t ∈ generatePiSystem (⋃ b, g b)
⊢ ∃ T f, t = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
state_after: case base
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
h_s : s ∈ ⋃ b, g b
⊢ ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b

case inter
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s t' : Set α
h_gen_s : generatePiSystem (⋃ b, g b) s
h_gen_t' : generatePiSystem (⋃ b, g b) t'
h_nonempty : (s ∩ t').Nonempty
h_s : ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
h_t' : ∃ T f, t' = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
⊢ ∃ T f, s ∩ t' = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b

tactic: rcases h_s with ⟨t', ⟨⟨b, rfl⟩, h_s_in_t'⟩⟩
state_before: case base
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
h_s : s ∈ ⋃ b, g b
⊢ ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
state_after: case base.intro.intro.intro
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
b : β
h_s_in_t' : s ∈ (fun b => g b) b
⊢ ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b

tactic: refine ⟨{b}, fun _ => s, ?_⟩
state_before: case base.intro.intro.intro
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
b : β
h_s_in_t' : s ∈ (fun b => g b) b
⊢ ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
state_after: case base.intro.intro.intro
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
b : β
h_s_in_t' : s ∈ (fun b => g b) b
⊢ s = ⋂ b_1 ∈ {b}, (fun x => s) b_1 ∧ ∀ b_1 ∈ {b}, (fun x => s) b_1 ∈ g b_1

tactic: simpa using h_s_in_t'
state_before: case base.intro.intro.intro
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
b : β
h_s_in_t' : s ∈ (fun b => g b) b
⊢ s = ⋂ b_1 ∈ {b}, (fun x => s) b_1 ∧ ∀ b_1 ∈ {b}, (fun x => s) b_1 ∈ g b_1
state_after: no goals

tactic: rcases h_t' with ⟨T_t', ⟨f_t', ⟨rfl, h_t'⟩⟩⟩
state_before: case inter
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s t' : Set α
h_gen_s : generatePiSystem (⋃ b, g b) s
h_gen_t' : generatePiSystem (⋃ b, g b) t'
h_nonempty : (s ∩ t').Nonempty
h_s : ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
h_t' : ∃ T f, t' = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
⊢ ∃ T f, s ∩ t' = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
state_after: case inter.intro.intro.intro
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
h_gen_s : generatePiSystem (⋃ b, g b) s
h_s : ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
h_nonempty : (s ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ ∃ T f, s ∩ ⋂ b ∈ T_t', f_t' b = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b

tactic: rcases h_s with ⟨T_s, ⟨f_s, ⟨rfl, h_s⟩⟩⟩
state_before: case inter.intro.intro.intro
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
h_gen_s : generatePiSystem (⋃ b, g b) s
h_s : ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
h_nonempty : (s ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ ∃ T f, s ∩ ⋂ b ∈ T_t', f_t' b = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
state_after: case inter.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ ∃ T f, (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b

tactic: use T_s ∪ T_t', fun b : β =>
  if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b
  else if b ∈ T_t' then f_t' b else (∅ : Set α)
state_before: case inter.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ ∃ T f, (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
state_after: case h
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b =
      ⋂ b ∈ T_s ∪ T_t',
        if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∧
    ∀ b ∈ T_s ∪ T_t',
      (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b

tactic: constructor
state_before: case h
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b =
      ⋂ b ∈ T_s ∪ T_t',
        if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∧
    ∀ b ∈ T_s ∪ T_t',
      (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b
state_after: case h.left
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b =
    ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅

case h.right
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ ∀ b ∈ T_s ∪ T_t',
    (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b

tactic: intro b h_b
state_before: case h.right
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ ∀ b ∈ T_s ∪ T_t',
    (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b
state_after: case h.right
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
⊢ (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b

tactic: split_ifs with hbs hbt hbt
state_before: case h.right
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
⊢ (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b
state_after: case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∈ T_s
hbt : b ∈ T_t'
⊢ f_s b ∩ f_t' b ∈ g b

case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∈ T_s
hbt : b ∉ T_t'
⊢ f_s b ∈ g b

case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∉ T_s
hbt : b ∈ T_t'
⊢ f_t' b ∈ g b

case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∉ T_s
hbt : b ∉ T_t'
⊢ ∅ ∈ g b

tactic: ext a
state_before: case h.left
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b =
    ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅
state_after: case h.left.h
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ a ∈ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ↔
    a ∈ ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅

tactic: simp_rw [Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union, or_imp]
state_before: case h.left.h
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ a ∈ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ↔
    a ∈ ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅
state_after: case h.left.h
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ ((∀ i ∈ T_s, a ∈ f_s i) ∧ ∀ i ∈ T_t', a ∈ f_t' i) ↔
    ∀ (i : β),
      (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧
        (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅)

tactic: rw [← forall_and]
state_before: case h.left.h
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ ((∀ i ∈ T_s, a ∈ f_s i) ∧ ∀ i ∈ T_t', a ∈ f_t' i) ↔
    ∀ (i : β),
      (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧
        (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅)
state_after: case h.left.h
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ (∀ (x : β), (x ∈ T_s → a ∈ f_s x) ∧ (x ∈ T_t' → a ∈ f_t' x)) ↔
    ∀ (i : β),
      (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧
        (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅)

tactic: constructor <;> intro h1 b <;> by_cases hbs : b ∈ T_s <;> by_cases hbt : b ∈ T_t' <;>
    specialize h1 b <;>
  simp only [hbs, hbt, if_true, if_false, true_imp_iff, and_self_iff, false_imp_iff,
    and_true_iff, true_and_iff] at h1 ⊢
state_before: case h.left.h
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ (∀ (x : β), (x ∈ T_s → a ∈ f_s x) ∧ (x ∈ T_t' → a ∈ f_t' x)) ↔
    ∀ (i : β),
      (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧
        (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅)
state_after: case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_s b ∧ a ∈ f_t' b
⊢ a ∈ f_s b ∩ f_t' b

case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∉ T_t'
h1 : a ∈ f_s b
⊢ a ∈ f_s b

case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∉ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_t' b
⊢ a ∈ f_t' b

case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_s b ∩ f_t' b
⊢ a ∈ f_s b ∧ a ∈ f_t' b

case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∉ T_t'
h1 : a ∈ f_s b
⊢ a ∈ f_s b

case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∉ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_t' b
⊢ a ∈ f_t' b

tactic: all_goals exact h1
state_before: case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_s b ∧ a ∈ f_t' b
⊢ a ∈ f_s b ∩ f_t' b

case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∉ T_t'
h1 : a ∈ f_s b
⊢ a ∈ f_s b

case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∉ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_t' b
⊢ a ∈ f_t' b

case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_s b ∩ f_t' b
⊢ a ∈ f_s b ∧ a ∈ f_t' b

case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∉ T_t'
h1 : a ∈ f_s b
⊢ a ∈ f_s b

case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∉ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_t' b
⊢ a ∈ f_t' b
state_after: no goals

tactic: exact h1
state_before: case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∉ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_t' b
⊢ a ∈ f_t' b
state_after: no goals

tactic: refine h_pi b (f_s b) (h_s b hbs) (f_t' b) (h_t' b hbt) (Set.Nonempty.mono ?_ h_nonempty)
state_before: case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∈ T_s
hbt : b ∈ T_t'
⊢ f_s b ∩ f_t' b ∈ g b
state_after: case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∈ T_s
hbt : b ∈ T_t'
⊢ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ⊆ f_s b ∩ f_t' b

tactic: exact Set.inter_subset_inter (Set.biInter_subset_of_mem hbs) (Set.biInter_subset_of_mem hbt)
state_before: case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∈ T_s
hbt : b ∈ T_t'
⊢ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ⊆ f_s b ∩ f_t' b
state_after: no goals

tactic: exact h_s b hbs
state_before: case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∈ T_s
hbt : b ∉ T_t'
⊢ f_s b ∈ g b
state_after: no goals

tactic: exact h_t' b hbt
state_before: case pos
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∉ T_s
hbt : b ∈ T_t'
⊢ f_t' b ∈ g b
state_after: no goals

tactic: rw [Finset.mem_union] at h_b
state_before: case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∉ T_s
hbt : b ∉ T_t'
⊢ ∅ ∈ g b
state_after: case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∨ b ∈ T_t'
hbs : b ∉ T_s
hbt : b ∉ T_t'
⊢ ∅ ∈ g b

tactic: apply False.elim (h_b.elim hbs hbt)
state_before: case neg
α : Type u_1
β : Type u_2
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∨ b ∈ T_t'
hbs : b ∉ T_s
hbt : b ∉ T_t'
⊢ ∅ ∈ g b
state_after: no goals