-- Mathlib/Topology/Compactness/Lindelof.lean
theorem isLindelof_of_countable_subfamily_closed
    (h :
      ∀ {ι : Type u} (t : ι → Set X), (∀ i, IsClosed (t i)) → (s ∩ ⋂ i, t i) = ∅ →
        ∃ u : Set ι, u.Countable ∧ (s ∩ ⋂ i ∈ u, t i) = ∅) :
    IsLindelof s :=
  isLindelof_of_countable_subcover fun U hUo hsU ↦ by
    rw [← disjoint_compl_right_iff_subset, compl_iUnion, disjoint_iff] at hsU
    rcases h (fun i ↦ (U i)ᶜ) (fun i ↦ (hUo _).isClosed_compl) hsU with ⟨t, ht⟩
    refine ⟨t, ?_⟩
    rwa [← disjoint_compl_right_iff_subset, compl_iUnion₂, disjoint_iff]

Proof state:
case intro
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h :
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
t : Set ι✝
ht : t.Countable ∧ s ∩ ⋂ i ∈ t, (U i)ᶜ = ∅
⊢ t.Countable ∧ s ⊆ ⋃ i ∈ t, U i

Tactic:
rwa [← disjoint_compl_right_iff_subset, compl_iUnion₂, disjoint_iff]

Result:
no goals

Proof state:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h :
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i

Tactic:
rw [← disjoint_compl_right_iff_subset, compl_iUnion, disjoint_iff] at hsU

Result:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h :
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i

Proof state:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h :
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i

Tactic:
rcases h (fun i ↦ (U i)ᶜ) (fun i ↦ (hUo _).isClosed_compl) hsU with ⟨t, ht⟩

Result:
case intro
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h :
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
t : Set ι✝
ht : t.Countable ∧ s ∩ ⋂ i ∈ t, (U i)ᶜ = ∅
⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i

Proof state:
case intro
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h :
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
t : Set ι✝
ht : t.Countable ∧ s ∩ ⋂ i ∈ t, (U i)ᶜ = ∅
⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i

Tactic:
refine ⟨t, ?_⟩

Result:
case intro
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h :
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed (t i)) → s ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen (U i)
hsU : s ⊓ ⋂ i, (U i)ᶜ = ⊥
t : Set ι✝
ht : t.Countable ∧ s ∩ ⋂ i ∈ t, (U i)ᶜ = ∅
⊢ t.Countable ∧ s ⊆ ⋃ i ∈ t, U i