Theorem: _private.Mathlib.Topology.ExtremallyDisconnected.0.ExtremallyDisconnected.homeoCompactToT2_injective
File: Mathlib/Topology/ExtremallyDisconnected.lean

Proof steps:
state_before: X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
⊢ Injective ρ
state_after: X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
⊢ x₁ = x₂
tactic: intro x₁ x₂ hρx
state_before: X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
⊢ x₁ = x₂
state_after: X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
⊢ False
tactic: by_contra hx
state_before: X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
⊢ False
state_after: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
⊢ False
tactic: rcases t2_separation hx with ⟨G₁, G₂, G₁_open, G₂_open, hx₁, hx₂, disj⟩
state_before: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
⊢ False
state_after: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
⊢ False
tactic: have G₁_comp : IsCompact G₁ᶜ := IsClosed.isCompact G₁_open.isClosed_compl
state_before: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
⊢ False
state_after: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
⊢ False
tactic: have G₂_comp : IsCompact G₂ᶜ := IsClosed.isCompact G₂_open.isClosed_compl
state_before: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
⊢ False
state_after: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
⊢ False
tactic: have G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ := (G₁_comp.image ρ_cont).isClosed.isOpen_compl
state_before: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
⊢ False
state_after: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
⊢ False
tactic: have G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ := (G₂_comp.image ρ_cont).isClosed.isOpen_compl
state_before: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
⊢ False
state_after: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ
⊢ False
tactic: have disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ := by
    rw [disjoint_iff_inter_eq_empty, ← compl_union, ← image_union, ← compl_inter,
      disjoint_iff_inter_eq_empty.mp disj, compl_empty, compl_empty_iff,
      image_univ_of_surjective ρ_surj]
state_before: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ
⊢ False
state_after: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ
disj'' : Disjoint (closure (ρ '' G₁ᶜ)ᶜ) (closure (ρ '' G₂ᶜ)ᶜ)
⊢ False
tactic: have disj'' : Disjoint (closure (ρ '' G₁ᶜ)ᶜ) (closure (ρ '' G₂ᶜ)ᶜ) :=
    disjoint_closure_of_disjoint_isOpen disj' G₁_open' G₂_open'
state_before: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ
disj'' : Disjoint (closure (ρ '' G₁ᶜ)ᶜ) (closure (ρ '' G₂ᶜ)ᶜ)
⊢ False
state_after: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ
disj'' : Disjoint (closure (ρ '' G₁ᶜ)ᶜ) (closure (ρ '' G₂ᶜ)ᶜ)
hx₁' : ρ x₁ ∈ closure (ρ '' G₁ᶜ)ᶜ
⊢ False
tactic: have hx₁' := image_subset_closure_compl_image_compl_of_isOpen ρ_cont ρ_surj zorn_subset G₁_open <|
    mem_image_of_mem ρ hx₁
state_before: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ
disj'' : Disjoint (closure (ρ '' G₁ᶜ)ᶜ) (closure (ρ '' G₂ᶜ)ᶜ)
hx₁' : ρ x₁ ∈ closure (ρ '' G₁ᶜ)ᶜ
⊢ False
state_after: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ
disj'' : Disjoint (closure (ρ '' G₁ᶜ)ᶜ) (closure (ρ '' G₂ᶜ)ᶜ)
hx₁' : ρ x₁ ∈ closure (ρ '' G₁ᶜ)ᶜ
hx₂' : ρ x₂ ∈ closure (ρ '' G₂ᶜ)ᶜ
⊢ False
tactic: have hx₂' := image_subset_closure_compl_image_compl_of_isOpen ρ_cont ρ_surj zorn_subset G₂_open <|
    mem_image_of_mem ρ hx₂
state_before: case intro.intro.intro.intro.intro.intro
X : Type u
inst✝⁷ : TopologicalSpace X
A D E : Type u
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace D
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ
G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ
disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ
disj'' : Disjoint (closure (ρ '' G₁ᶜ)ᶜ) (closure (ρ '' G₂ᶜ)ᶜ)
hx₁' : ρ x₁ ∈ closure (ρ '' G₁ᶜ)ᶜ
hx₂' : ρ x₂ ∈ closure (ρ '' G₂ᶜ)ᶜ
⊢ False
state_after: no goals
tactic: exact disj''.ne_of_mem hx₁' hx₂' hρx