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