-- Mathlib/Topology/Separation/Basic.lean
theorem t1Space_TFAE (X : Type u) [TopologicalSpace X] :
    List.TFAE [T1Space X,
      ∀ x, IsClosed ({ x } : Set X),
      ∀ x, IsOpen ({ x }ᶜ : Set X),
      Continuous (@CofiniteTopology.of X),
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U,
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
      ∀ ⦃x y : X⦄, x ⤳ y → x = y]

Proof state:
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed {x}) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_6 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s
tfae_5_iff_7 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U
tfae_5_iff_8 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE

Tactic:
tfae_have 8 ↔ 9 := forall_swap.trans (by simp only [disjoint_comm, ne_comm])

Result:
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed {x}) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_6 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s
tfae_5_iff_7 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U
tfae_5_iff_8 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
tfae_8_iff_9 : (∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y)
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE

Proof state:
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed {x}) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen {x}ᶜ
⊢ (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s

Tactic:
simp only [← subset_compl_singleton_iff, exists_mem_subset_iff]

Result:
no goals

Proof state:
X : Type u
inst✝ : TopologicalSpace X
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE

Tactic:
tfae_have 1 ↔ 2 := ⟨fun h => h.1, fun h => ⟨h⟩⟩

Result:
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE

Proof state:
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed {x}) ↔ ∀ (x : X), IsOpen {x}ᶜ
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE

Tactic:
tfae_have 5 ↔ 3 := by
  refine forall_swap.trans ?_
  simp only [isOpen_iff_mem_nhds, mem_compl_iff, mem_singleton_iff]

Result:
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed {x}) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen {x}ᶜ
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE

Proof state:
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed {x}) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_6 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s
tfae_5_iff_7 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U
tfae_5_iff_8 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
tfae_8_iff_9 : (∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y)
tfae_1_to_4 : T1Space X → Continuous ⇑CofiniteTopology.of
tfae_4_to_2 : Continuous ⇑CofiniteTopology.of → ∀ (x : X), IsClosed {x}
⊢ (∀ (x : X), IsClosed {x}) ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y

Tactic:
simp only [← closure_subset_iff_isClosed, specializes_iff_mem_closure, subset_def,
  mem_singleton_iff, eq_comm]

Result:
no goals