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

State before:
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
⊢ Set.empty.union s = s
Tactic:
apply Set.ext
State after:
case a
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
⊢ ∀ (x : α), x ∈ Set.empty.union s ↔ x ∈ s

State before:
case a
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
⊢ ∀ (x : α), x ∈ Set.empty.union s ↔ x ∈ s
Tactic:
intro x
State after:
case a
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
⊢ x ∈ Set.empty.union s ↔ x ∈ s

State before:
case a
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
⊢ x ∈ Set.empty.union s ↔ x ∈ s
Tactic:
constructor
State after:
case a.mp
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
⊢ x ∈ Set.empty.union s → x ∈ s

case a.mpr
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
⊢ x ∈ s → x ∈ Set.empty.union s

State before:
case a.mp
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
⊢ x ∈ Set.empty.union s → x ∈ s
Tactic:
intro
| Or.inl _ => contradiction
| Or.inr _ => assumption
State after:
no goals

State before:
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
x✝ : x ∈ Set.empty.union s
h✝ : x ∈ Set.empty
⊢ x ∈ s
Tactic:
contradiction
State after:
no goals

State before:
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
x✝ : x ∈ Set.empty.union s
h✝ : x ∈ s
⊢ x ∈ s
Tactic:
assumption
State after:
no goals

State before:
case a.mpr
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
⊢ x ∈ s → x ∈ Set.empty.union s
Tactic:
intro h
State after:
case a.mpr
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
h : x ∈ s
⊢ x ∈ Set.empty.union s

State before:
case a.mpr
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
h : x ∈ s
⊢ x ∈ Set.empty.union s
Tactic:
right
State after:
case a.mpr.h
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
h : x ∈ s
⊢ x ∈ s

State before:
case a.mpr.h
α : Sort u_1
β : ?m.13016
γ : ?m.13019
s : Set α
x : α
h : x ∈ s
⊢ x ∈ s
Tactic:
exact h
State after:
no goals