-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
neg neg₁ neg₂ neg₃ : Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₄ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₅ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S) (h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)),
    Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₆ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S) (h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)) (n : ℕ),
    (dissipate s' n).Nonempty
neg₇ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S) (h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)) (j : ℕ)
    (hj : dissipate s' j = ∅), False
neg₈ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S) (h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)) (j : ℕ)
    (hj : dissipate s' j = ∅) (h₃ : ∀ (v : ℕ) (hv : n ≤ v), dissipate s v = dissipate s' v), False
neg₉ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S) (h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)) (j : ℕ)
    (hj : dissipate s' j = ∅) (h₃ : ∀ (v : ℕ) (hv : n ≤ v), dissipate s v = dissipate s' v)
    (h₇ : dissipate s' (max j n) = ∅), False
neg₁₀ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S) (h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)) (j : ℕ)
    (hj : dissipate s' j = ∅) (h₇ : dissipate s' (max j n) = ∅) (h₃ : dissipate s (max j n) = dissipate s' (max j n)),
    False
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
have neg₁₁ : let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
∀ (h₁ : ∀ (i : ℕ), s' i ∈ S) (h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)) (j : ℕ)
  (hj : dissipate s' j = ∅) (h₇ : dissipate s' (max j n) = ∅) (h₃ : dissipate s (max j n) = dissipate s' (max j n))
  (hd : (dissipate s (max j n)).Nonempty), False
Tactic: classical
  let n := Nat.find h₀
  let s' := fun i ↦ if s i ∈ S then s i else s n
  have h₁ : ∀ i, s' i ∈ S := by grind
  have h₂ : ⋂ i, s i = ⋂ i, s' i := by ext; simp; grind
  apply h₂ ▸ h s' h₁
  by_contra! ⟨j, hj⟩
  have h₃ (v : ℕ) (hv : n ≤ v) : dissipate s v = dissipate s' v := by ext; simp; grind
  have h₇ : dissipate s' (max j n) = ∅ := by
    rw [← subset_empty_iff] at hj ⊢
    exact (antitone_dissipate (Nat.le_max_left j n)).trans hj
  specialize h₃ (max j n) (Nat.le_max_right j n)
  specialize hd (max j n)
  simp [h₃, h₇] at hd
Cleared hypotheses: hd