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

State before:
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
⊢ (Set.pure a).inter s = Set.pureIf a (a ∈ s)
Tactic:
apply Set.ext
State after:
case a
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
⊢ ∀ (x : α), x ∈ (Set.pure a).inter s ↔ x ∈ Set.pureIf a (a ∈ s)

State before:
case a
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
⊢ ∀ (x : α), x ∈ (Set.pure a).inter s ↔ x ∈ Set.pureIf a (a ∈ s)
Tactic:
intro x
State after:
case a
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
x : α
⊢ x ∈ (Set.pure a).inter s ↔ x ∈ Set.pureIf a (a ∈ s)

State before:
case a
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
x : α
⊢ x ∈ (Set.pure a).inter s ↔ x ∈ Set.pureIf a (a ∈ s)
Tactic:
constructor
State after:
case a.mp
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
x : α
⊢ x ∈ (Set.pure a).inter s → x ∈ Set.pureIf a (a ∈ s)

case a.mpr
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
x : α
⊢ x ∈ Set.pureIf a (a ∈ s) → x ∈ (Set.pure a).inter s

State before:
case a.mp
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
x : α
⊢ x ∈ (Set.pure a).inter s → x ∈ Set.pureIf a (a ∈ s)
Tactic:
intro ⟨rfl, hs⟩
State after:
case a.mp
α : Sort u_1
β : ?m.14390
γ : ?m.14393
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.14390
γ : ?m.14393
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.14390
γ : ?m.14393
a : α
s : Set α
x : α
⊢ x ∈ Set.pureIf a (a ∈ s) → x ∈ (Set.pure a).inter s
Tactic:
intro ⟨rfl, hs⟩
State after:
case a.mpr
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
x : α
hs : x ∈ s
⊢ x ∈ (Set.pure x).inter s

State before:
case a.mpr
α : Sort u_1
β : ?m.14390
γ : ?m.14393
a : α
s : Set α
x : α
hs : x ∈ s
⊢ x ∈ (Set.pure x).inter s
Tactic:
refine ⟨rfl, hs⟩
State after:
no goals