∀ {α : Type u_1} [inst : DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} (motive : (∃ u ∈ 𝒜, s ∩ u = s) → Prop)
  (x : ∃ u ∈ 𝒜, s ∩ u = s), (∀ (t : Finset α) (ht : t ∈ 𝒜) (hst : s ∩ t = s), motive ⋯) → motive x