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

Proof steps:
state_before: X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
state_after: case tfae_1_iff_5
X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
⊢ RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x

X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
tactic: tfae_have 1 ↔ 5
state_before: X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
state_after: case tfae_5_to_6
X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
⊢ (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x

X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
tactic: tfae_have 5 → 6
state_before: X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
state_after: case tfae_6_to_4
X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
⊢ (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s

X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
tactic: tfae_have 6 → 4
state_before: X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
state_after: case tfae_4_to_2
X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
⊢ (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) → ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)

X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) → ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
tactic: tfae_have 4 → 2
state_before: X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) → ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
state_after: case tfae_2_to_3
X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) → ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)
⊢ (∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)) → ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s

X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) → ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)
tfae_2_to_3 :
  (∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)) → ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
tactic: tfae_have 2 → 3
state_before: X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) → ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)
tfae_2_to_3 :
  (∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)) → ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
state_after: case tfae_3_to_1
X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) → ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)
tfae_2_to_3 :
  (∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)) → ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s
⊢ (∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s) → RegularSpace X

X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) → ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)
tfae_2_to_3 :
  (∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)) → ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s
tfae_3_to_1 : (∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s) → RegularSpace X
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
tactic: tfae_have 3 → 1
state_before: X✝ : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X✝
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_5 : RegularSpace X ↔ ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x
tfae_5_to_6 : (∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x) → ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x
tfae_6_to_4 : (∀ (x : X), (𝓝 x).lift' closure = 𝓝 x) → ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
tfae_4_to_2 :
  (∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) → ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)
tfae_2_to_3 :
  (∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x)) → ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s
tfae_3_to_1 : (∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s) → RegularSpace X
⊢ [RegularSpace X, ∀ (s : Set X), ∀ x ∉ closure s, Disjoint (𝓝ˢ s) (𝓝 x),
      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s, ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
      ∀ (x : X), (𝓝 x).lift' closure ≤ 𝓝 x, ∀ (x : X), (𝓝 x).lift' closure = 𝓝 x].TFAE
state_after: no goals
tactic: tfae_finish