∀ {R : Type u_1} {K : Type u_2} [inst : CommRing R] [inst_1 : Field K] [inst_2 : Algebra R K] [IsFractionRing R K]
  {ι : Type u_3} (S : ι → Subalgebra R K), (∀ (i : ι), IsIntegrallyClosed ↥(S i)) → ⨅ i, S i = ⊥ → IsIntegrallyClosed R