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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(* A Type-theoretical axiom of choice gives us the collection axiom  *)
Definition collection :=
forall P : Ens -> Ens -> Prop,
(forall x x' y : Ens, EQ x x' -> P x y -> P x' y) ->
(forall E : Ens, EXType _ (P E)) ->
forall E : Ens,
EXType _
(fun A : Ens =>
forall x : Ens, IN x E -> EXType _ (fun y : Ens => IN y A /\ P x y)).
Definition choice :=
forall (A B : Type) (P : A -> B -> Prop),
(forall a : A, EXType _ (fun b : B => P a b)) ->
EXType _ (fun f : A -> B => forall a : A, P a (f a)).

(* Specifications: *)
Theorem Choice_Collection : choice -> collection.
unfold collection in |- *; intro; intros P comp G E.