Definition subset (U:Type) (P Q:U->Prop) : Prop := forall x, P x -> Q x.

(* Specifications: *)
Theorem singleton_choice : forall (A : Type) (P : A->Prop), (exists x : A, P x) -> exists P' : A->Prop, subset P' P /\ exists! x, P' x.
Theorem choice : forall (A B : Type) (R : A->B->Prop), (forall x : A, exists y : B, R x y) -> exists f : A->B, (forall x : A, R x (f x)).