Theorem: TopologicalSpace.IsTopologicalBasis.diff_empty
File: Mathlib/Topology/Bases.lean

tactic: refine ⟨?_, by rw [sUnion_diff_singleton_empty, h.sUnion_eq], ?_⟩
state_before: α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ IsTopologicalBasis (s \ {∅})
state_after: case refine_1
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ ∀ t₁ ∈ s \ {∅}, ∀ t₂ ∈ s \ {∅}, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s \ {∅}, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂

case refine_2
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ t = generateFrom (s \ {∅})

tactic: rw [sUnion_diff_singleton_empty, h.sUnion_eq]
state_before: α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ ⋃₀ (s \ {∅}) = univ
state_after: no goals

tactic: rintro t₁ ⟨h₁, -⟩ t₂ ⟨h₂, -⟩ x hx
state_before: case refine_1
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ ∀ t₁ ∈ s \ {∅}, ∀ t₂ ∈ s \ {∅}, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s \ {∅}, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
state_after: case refine_1.intro.intro
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
t₁ : Set α
h₁ : t₁ ∈ s
t₂ : Set α
h₂ : t₂ ∈ s
x : α
hx : x ∈ t₁ ∩ t₂
⊢ ∃ t₃ ∈ s \ {∅}, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂

tactic: obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x hx
state_before: case refine_1.intro.intro
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
t₁ : Set α
h₁ : t₁ ∈ s
t₂ : Set α
h₂ : t₂ ∈ s
x : α
hx : x ∈ t₁ ∩ t₂
⊢ ∃ t₃ ∈ s \ {∅}, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
state_after: case refine_1.intro.intro.intro.intro
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
t₁ : Set α
h₁ : t₁ ∈ s
t₂ : Set α
h₂ : t₂ ∈ s
x : α
hx : x ∈ t₁ ∩ t₂
t₃ : Set α
h₃ : t₃ ∈ s
hs : x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
⊢ ∃ t₃ ∈ s \ {∅}, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂

tactic: exact ⟨t₃, ⟨h₃, Nonempty.ne_empty ⟨x, hs.1⟩⟩, hs⟩
state_before: case refine_1.intro.intro.intro.intro
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
t₁ : Set α
h₁ : t₁ ∈ s
t₂ : Set α
h₂ : t₂ ∈ s
x : α
hx : x ∈ t₁ ∩ t₂
t₃ : Set α
h₃ : t₃ ∈ s
hs : x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
⊢ ∃ t₃ ∈ s \ {∅}, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
state_after: no goals

tactic: rw [h.eq_generateFrom]
state_before: case refine_2
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ t = generateFrom (s \ {∅})
state_after: case refine_2
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ generateFrom s = generateFrom (s \ {∅})

tactic: refine le_antisymm (generateFrom_anti diff_subset) (le_generateFrom fun t ht => ?_)
state_before: case refine_2
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ generateFrom s = generateFrom (s \ {∅})
state_after: case refine_2
α : Type u
β : Type u_1
t✝ : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
t : Set α
ht : t ∈ s
⊢ IsOpen t

tactic: obtain rfl | he := eq_or_ne t ∅
state_before: case refine_2
α : Type u
β : Type u_1
t✝ : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
t : Set α
ht : t ∈ s
⊢ IsOpen t
state_after: case refine_2.inl
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
ht : ∅ ∈ s
⊢ IsOpen ∅

case refine_2.inr
α : Type u
β : Type u_1
t✝ : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
t : Set α
ht : t ∈ s
he : t ≠ ∅
⊢ IsOpen t

tactic: exact @isOpen_empty _ (generateFrom _)
state_before: case refine_2.inl
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
ht : ∅ ∈ s
⊢ IsOpen ∅
state_after: no goals

tactic: exact .basic t ⟨ht, he⟩
state_before: case refine_2.inr
α : Type u
β : Type u_1
t✝ : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
h : IsTopologicalBasis s
t : Set α
ht : t ∈ s
he : t ≠ ∅
⊢ IsOpen t
state_after: no goals