intros x z H'; elim H'; auto with sets.
Qed.
Hint Resolve Upper_Bound_Singleton_inv.

(* Specifications: *)
Theorem Non_empty_has_glb :
Conditionally_complete U D ->
forall X : Ensemble U,
Included U X (Carrier_of U D) ->
Inhabited U X -> exists binf : U, Glb U D X binf.
Theorem Bounded_implies_consistent :
forall X : Ensemble U,
Included U X (Carrier_of U D) ->
ex (fun maj : U => Upper_Bound U D X maj) -> Consistent U D X.