∀ {X : Type u} [inst : TopologicalSpace X] {S : Set (Set X)} [hS : Nonempty ↑S],
  DirectedOn (fun x1 x2 => x1 ⊇ x2) S →
    (∀ U ∈ S, U.Nonempty) → (∀ U ∈ S, IsCompact U) → (∀ U ∈ S, IsClosed U) → (⋂₀ S).Nonempty