∀ {α : Type u_1} [inst : CommSemiring α] (I : Ideal α) (S : Submonoid α)
  (motive : (∃ m, I ≤ m ∧ Maximal (fun x => x ∈ {p | Disjoint ↑p ↑S}) m) → Prop)
  (x : ∃ m, I ≤ m ∧ Maximal (fun x => x ∈ {p | Disjoint ↑p ↑S}) m),
  (∀ (p : Ideal α) (hIp : I ≤ p) (hp : Maximal (fun x => x ∈ {p | Disjoint ↑p ↑S}) p), motive ⋯) → motive x