Record nested_IN (A B C : Ens) : Prop :=
{nested_IN_ab : IN A B; nested_IN_bc : IN B C}.
Definition b_in_Union_a (A B : Ens) :=
EXISTS (fun C : Ens => nested_IN B C A).

(******** singletons ****************)
Axiom Singleton_pr1 : forall X : Ens, IN X (Singleton X).