Theorem: LO.FirstOrder.Arith.Model.sUnion_exists_unique (Arithmetization/ISigmaOne/HFS/Basic.lean)

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
s : M
⊢ ∃! u, ∀ (x : M), x ∈ u ↔ ∃ t ∈ s, x ∈ t
Tactic:
have : 𝚺₁-Predicate fun x ↦ ∃ t ∈ s, x ∈ t := by definability
State after:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
s : M
this : 𝚺₁-Predicate fun x => ∃ t ∈ s, x ∈ t
⊢ ∃! u, ∀ (x : M), x ∈ u ↔ ∃ t ∈ s, x ∈ t

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
s : M
this : 𝚺₁-Predicate fun x => ∃ t ∈ s, x ∈ t
⊢ ∃! u, ∀ (x : M), x ∈ u ↔ ∃ t ∈ s, x ∈ t
Tactic:
exact finite_comprehension₁! this
  ⟨s, fun i ↦ by
    rintro ⟨t, ht, hi⟩; exact lt_trans _ _ _ (lt_of_mem hi) (lt_of_mem ht)⟩
State after:
no goals

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
s : M
⊢ 𝚺₁-Predicate fun x => ∃ t ∈ s, x ∈ t
Tactic:
definability
State after:
no goals

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
s : M
this : 𝚺₁-Predicate fun x => ∃ t ∈ s, x ∈ t
i : M
⊢ (∃ t ∈ s, i ∈ t) → i < s
Tactic:
rintro ⟨t, ht, hi⟩
State after:
case intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
s : M
this : 𝚺₁-Predicate fun x => ∃ t ∈ s, x ∈ t
i t : M
ht : t ∈ s
hi : i ∈ t
⊢ i < s

State before:
case intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
s : M
this : 𝚺₁-Predicate fun x => ∃ t ∈ s, x ∈ t
i t : M
ht : t ∈ s
hi : i ∈ t
⊢ i < s
Tactic:
exact lt_trans _ _ _ (lt_of_mem hi) (lt_of_mem ht)
State after:
no goals