Theorem: MeasurableSpace.DynkinSystem.ext
File: Mathlib/MeasureTheory/PiSystem.lean

Proof steps:
state_before: α : Type u_1
s₁ : Set α → Prop
has_empty✝¹ : s₁ ∅
has_compl✝¹ : ∀ {a : Set α}, s₁ a → s₁ aᶜ
has_iUnion_nat✝¹ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i)
s₂ : Set α → Prop
has_empty✝ : s₂ ∅
has_compl✝ : ∀ {a : Set α}, s₂ a → s₂ aᶜ
has_iUnion_nat✝ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₂ (f i)) → s₂ (⋃ i, f i)
h :
  ∀ (s : Set α),
    { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ }.Has s ↔
      { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }.Has s
⊢ { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } =
    { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }
state_after: α : Type u_1
s₁ : Set α → Prop
has_empty✝¹ : s₁ ∅
has_compl✝¹ : ∀ {a : Set α}, s₁ a → s₁ aᶜ
has_iUnion_nat✝¹ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i)
s₂ : Set α → Prop
has_empty✝ : s₂ ∅
has_compl✝ : ∀ {a : Set α}, s₂ a → s₂ aᶜ
has_iUnion_nat✝ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₂ (f i)) → s₂ (⋃ i, f i)
h :
  ∀ (s : Set α),
    { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ }.Has s ↔
      { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }.Has s
this : s₁ = s₂
⊢ { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } =
    { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }
tactic: have : s₁ = s₂ := funext fun x => propext <| h x
state_before: α : Type u_1
s₁ : Set α → Prop
has_empty✝¹ : s₁ ∅
has_compl✝¹ : ∀ {a : Set α}, s₁ a → s₁ aᶜ
has_iUnion_nat✝¹ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i)
s₂ : Set α → Prop
has_empty✝ : s₂ ∅
has_compl✝ : ∀ {a : Set α}, s₂ a → s₂ aᶜ
has_iUnion_nat✝ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₂ (f i)) → s₂ (⋃ i, f i)
h :
  ∀ (s : Set α),
    { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ }.Has s ↔
      { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }.Has s
this : s₁ = s₂
⊢ { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } =
    { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }
state_after: α : Type u_1
s₁ : Set α → Prop
has_empty✝¹ : s₁ ∅
has_compl✝¹ : ∀ {a : Set α}, s₁ a → s₁ aᶜ
has_iUnion_nat✝¹ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i)
has_empty✝ : s₁ ∅
has_compl✝ : ∀ {a : Set α}, s₁ a → s₁ aᶜ
has_iUnion_nat✝ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i)
h :
  ∀ (s : Set α),
    { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ }.Has s ↔
      { Has := s₁, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }.Has s
⊢ { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } =
    { Has := s₁, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }
tactic: subst this
state_before: α : Type u_1
s₁ : Set α → Prop
has_empty✝¹ : s₁ ∅
has_compl✝¹ : ∀ {a : Set α}, s₁ a → s₁ aᶜ
has_iUnion_nat✝¹ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i)
has_empty✝ : s₁ ∅
has_compl✝ : ∀ {a : Set α}, s₁ a → s₁ aᶜ
has_iUnion_nat✝ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i)
h :
  ∀ (s : Set α),
    { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ }.Has s ↔
      { Has := s₁, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }.Has s
⊢ { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } =
    { Has := s₁, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ }
state_after: no goals
tactic: rfl