∀ {R : Type u_1} [inst : CommSemiring R] (s : Set R) (x y : R)
  (motive : (∃ r ∈ s, Disjoint ↑(Module.eqIdeal R x y) ↑(Submonoid.powers r)) → Prop)
  (x_1 : ∃ r ∈ s, Disjoint ↑(Module.eqIdeal R x y) ↑(Submonoid.powers r)),
  (∀ (r : R) (hrs : r ∈ s) (disj : Disjoint ↑(Module.eqIdeal R x y) ↑(Submonoid.powers r)), motive ⋯) → motive x_1