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

State before:
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
⊢ f.seq Set.empty = Set.empty
Tactic:
apply Set.ext
State after:
case a
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
⊢ ∀ (x : β), x ∈ f.seq Set.empty ↔ x ∈ Set.empty

State before:
case a
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
⊢ ∀ (x : β), x ∈ f.seq Set.empty ↔ x ∈ Set.empty
Tactic:
intro x
State after:
case a
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
⊢ x ∈ f.seq Set.empty ↔ x ∈ Set.empty

State before:
case a
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
⊢ x ∈ f.seq Set.empty ↔ x ∈ Set.empty
Tactic:
constructor
State after:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
⊢ x ∈ f.seq Set.empty → x ∈ Set.empty

case a.mpr
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
⊢ x ∈ Set.empty → x ∈ f.seq Set.empty

State before:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
⊢ x ∈ f.seq Set.empty → x ∈ Set.empty
Tactic:
intro ⟨_, _, (hx : x ∈ Set.empty.map _)⟩
State after:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
w✝ : α → β
left✝ : w✝ ∈ f
hx : x ∈ Set.map w✝ Set.empty
⊢ x ∈ Set.empty

State before:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
w✝ : α → β
left✝ : w✝ ∈ f
hx : x ∈ Set.map w✝ Set.empty
⊢ x ∈ Set.empty
Tactic:
rw [empty_map] at hx
State after:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
w✝ : α → β
left✝ : w✝ ∈ f
hx : x ∈ Set.empty
⊢ x ∈ Set.empty

State before:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
w✝ : α → β
left✝ : w✝ ∈ f
hx : x ∈ Set.empty
⊢ x ∈ Set.empty
Tactic:
contradiction
State after:
no goals

State before:
case a.mpr
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
⊢ x ∈ Set.empty → x ∈ f.seq Set.empty
Tactic:
intro _
State after:
case a.mpr
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
a✝ : x ∈ Set.empty
⊢ x ∈ f.seq Set.empty

State before:
case a.mpr
α : Sort u_1
β : Sort u_2
γ : ?m.14047
f : Set (α → β)
x : β
a✝ : x ∈ Set.empty
⊢ x ∈ f.seq Set.empty
Tactic:
contradiction
State after:
no goals