intros x z H' H'0 H'1; rewrite <- H'1.
apply Upper_Bound_definition; auto with sets.
intros y H'2; elim H'2; auto with sets.
Qed.
Hint Resolve Upper_Bound_Singleton_intro.

Theorem Upper_Bound_Singleton_inv :
forall x z : U, Upper_Bound U D (Singleton U x) z -> Rel_of U D x z.