intros x H'; auto 20.
apply Lub_definition; auto 10.
apply Upper_Bound_definition; auto 10.
intros y H'0; elim H'0; auto 10.
intros y H'0; elim H'0; auto 10.
Qed.
Hint Resolve Singleton_has_lub.

Theorem Empty_set_has_Upper_Bound :
exists maj : U, Upper_Bound U D (Empty_set U) maj.