Section The_power_set_partial_order.
Variable U : Type.
Inductive Power_set (A : Ensemble U) : Ensemble (Ensemble U) :=
Definition_of_Power_set :
forall X : Ensemble U,
Included U X A -> In (Ensemble U) (Power_set A) X.
Hint Resolve Definition_of_Power_set.
(* Consider a set A, and its powerset P  *)
Variable A : Ensemble U.
Hint Resolve Empty_set_minimal.
Hint Resolve Power_set_non_empty.
Hint Resolve Inclusion_is_an_order.
elim Inclusion_is_an_order; auto.
Qed.
Hint Resolve Inclusion_is_transitive.
Hint Resolve Same_set_reflexive.
Hint Resolve Union_minimal.
Hint Resolve Union_increases_l Union_increases_r Intersection_decreases_l
Intersection_decreases_r.
Hint Resolve Empty_set_is_Bottom.

Theorem Non_disjoint_union :
forall (X : Ensemble U) (x : U),
In U X x -> Union U (Singleton U x) X = X.