Theorem: disjoint_nested_nhds
File: Mathlib/Topology/Separation.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T3Space X
x y : X
h : x ≠ y
⊢ ∃ U₁ ∈ 𝓝 x,
    ∃ V₁ ∈ 𝓝 x,
      ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y, IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂
state_after: case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T3Space X
x y : X
h : x ≠ y
U₁ U₂ : Set X
U₁_op : IsOpen U₁
U₂_op : IsOpen U₂
x_in : x ∈ U₁
y_in : y ∈ U₂
H : Disjoint U₁ U₂
⊢ ∃ U₁ ∈ 𝓝 x,
    ∃ V₁ ∈ 𝓝 x,
      ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y, IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂
tactic: rcases t2_separation h with ⟨U₁, U₂, U₁_op, U₂_op, x_in, y_in, H⟩
state_before: case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T3Space X
x y : X
h : x ≠ y
U₁ U₂ : Set X
U₁_op : IsOpen U₁
U₂_op : IsOpen U₂
x_in : x ∈ U₁
y_in : y ∈ U₂
H : Disjoint U₁ U₂
⊢ ∃ U₁ ∈ 𝓝 x,
    ∃ V₁ ∈ 𝓝 x,
      ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y, IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T3Space X
x y : X
h : x ≠ y
U₁ U₂ : Set X
U₁_op : IsOpen U₁
U₂_op : IsOpen U₂
x_in : x ∈ U₁
y_in : y ∈ U₂
H : Disjoint U₁ U₂
V₁ : Set X
V₁_in : V₁ ∈ 𝓝 x
V₁_closed : IsClosed V₁
h₁ : V₁ ⊆ U₁
⊢ ∃ U₁ ∈ 𝓝 x,
    ∃ V₁ ∈ 𝓝 x,
      ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y, IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂
tactic: rcases exists_mem_nhds_isClosed_subset (U₁_op.mem_nhds x_in) with ⟨V₁, V₁_in, V₁_closed, h₁⟩
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T3Space X
x y : X
h : x ≠ y
U₁ U₂ : Set X
U₁_op : IsOpen U₁
U₂_op : IsOpen U₂
x_in : x ∈ U₁
y_in : y ∈ U₂
H : Disjoint U₁ U₂
V₁ : Set X
V₁_in : V₁ ∈ 𝓝 x
V₁_closed : IsClosed V₁
h₁ : V₁ ⊆ U₁
⊢ ∃ U₁ ∈ 𝓝 x,
    ∃ V₁ ∈ 𝓝 x,
      ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y, IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T3Space X
x y : X
h : x ≠ y
U₁ U₂ : Set X
U₁_op : IsOpen U₁
U₂_op : IsOpen U₂
x_in : x ∈ U₁
y_in : y ∈ U₂
H : Disjoint U₁ U₂
V₁ : Set X
V₁_in : V₁ ∈ 𝓝 x
V₁_closed : IsClosed V₁
h₁ : V₁ ⊆ U₁
V₂ : Set X
V₂_in : V₂ ∈ 𝓝 y
V₂_closed : IsClosed V₂
h₂ : V₂ ⊆ U₂
⊢ ∃ U₁ ∈ 𝓝 x,
    ∃ V₁ ∈ 𝓝 x,
      ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y, IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂
tactic: rcases exists_mem_nhds_isClosed_subset (U₂_op.mem_nhds y_in) with ⟨V₂, V₂_in, V₂_closed, h₂⟩
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T3Space X
x y : X
h : x ≠ y
U₁ U₂ : Set X
U₁_op : IsOpen U₁
U₂_op : IsOpen U₂
x_in : x ∈ U₁
y_in : y ∈ U₂
H : Disjoint U₁ U₂
V₁ : Set X
V₁_in : V₁ ∈ 𝓝 x
V₁_closed : IsClosed V₁
h₁ : V₁ ⊆ U₁
V₂ : Set X
V₂_in : V₂ ∈ 𝓝 y
V₂_closed : IsClosed V₂
h₂ : V₂ ⊆ U₂
⊢ ∃ U₁ ∈ 𝓝 x,
    ∃ V₁ ∈ 𝓝 x,
      ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y, IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂
state_after: no goals
tactic: exact ⟨U₁, mem_of_superset V₁_in h₁, V₁, V₁_in, U₂, mem_of_superset V₂_in h₂, V₂, V₂_in,
    V₁_closed, V₂_closed, U₁_op, U₂_op, h₁, h₂, H⟩