Theorem: Set.inter_pure (GMLInit/Data/Set/Basic.lean)

State before:
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
⊢ s.inter (Set.pure a) = Set.pureIf a (a ∈ s)
Tactic:
apply Set.ext
State after:
case a
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
⊢ ∀ (x : α), x ∈ s.inter (Set.pure a) ↔ x ∈ Set.pureIf a (a ∈ s)

State before:
case a
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
⊢ ∀ (x : α), x ∈ s.inter (Set.pure a) ↔ x ∈ Set.pureIf a (a ∈ s)
Tactic:
intro x
State after:
case a
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
⊢ x ∈ s.inter (Set.pure a) ↔ x ∈ Set.pureIf a (a ∈ s)

State before:
case a
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
⊢ x ∈ s.inter (Set.pure a) ↔ x ∈ Set.pureIf a (a ∈ s)
Tactic:
constructor
State after:
case a.mp
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
⊢ x ∈ s.inter (Set.pure a) → x ∈ Set.pureIf a (a ∈ s)

case a.mpr
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
⊢ x ∈ Set.pureIf a (a ∈ s) → x ∈ s.inter (Set.pure a)

State before:
case a.mp
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
⊢ x ∈ s.inter (Set.pure a) → x ∈ Set.pureIf a (a ∈ s)
Tactic:
intro ⟨hs, rfl⟩
State after:
case a.mp
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
hs : x ∈ s
⊢ x ∈ Set.pureIf x (x ∈ s)

State before:
case a.mp
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
hs : x ∈ s
⊢ x ∈ Set.pureIf x (x ∈ s)
Tactic:
refine ⟨rfl, hs⟩
State after:
no goals

State before:
case a.mpr
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
⊢ x ∈ Set.pureIf a (a ∈ s) → x ∈ s.inter (Set.pure a)
Tactic:
intro ⟨rfl, hs⟩
State after:
case a.mpr
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
hs : x ∈ s
⊢ x ∈ s.inter (Set.pure x)

State before:
case a.mpr
α : Sort u_1
β : ?m.15184
γ : ?m.15187
a : α
s : Set α
x : α
hs : x ∈ s
⊢ x ∈ s.inter (Set.pure x)
Tactic:
refine ⟨hs, rfl⟩
State after:
no goals