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.

Theorem Power_set_non_empty :
forall A : Ensemble U, Non_empty (Ensemble U) (Power_set A).