# Library Coq.Sets.Constructive_sets

Require Export Ensembles.

Section Ensembles_facts.
Variable U : Type.

Lemma Extension : forall B C:Ensemble U, B = C -> Same_set U B C.

Lemma Noone_in_empty : forall x:U, ~ In U (Empty_set U) x.

Lemma Included_Empty : forall A:Ensemble U, Included U (Empty_set U) A.

forall (A:Ensemble U) (x y:U), In U A y -> In U (Add U A x) y.

Lemma Add_intro2 : forall (A:Ensemble U) (x:U), In U (Add U A x) x.

Lemma Inhabited_add : forall (A:Ensemble U) (x:U), Inhabited U (Add U A x).

Lemma Inhabited_not_empty :
forall X:Ensemble U, Inhabited U X -> X <> Empty_set U.

Lemma Add_not_Empty : forall (A:Ensemble U) (x:U), Add U A x <> Empty_set U.

Lemma not_Empty_Add : forall (A:Ensemble U) (x:U), Empty_set U <> Add U A x.

Lemma Singleton_inv : forall x y:U, In U (Singleton U x) y -> x = y.

Lemma Singleton_intro : forall x y:U, x = y -> In U (Singleton U x) y.

Lemma Union_inv :
forall (B C:Ensemble U) (x:U), In U (Union U B C) x -> In U B x \/ In U C x.

forall (A:Ensemble U) (x y:U), In U (Add U A x) y -> In U A y \/ x = y.

Lemma Intersection_inv :
forall (B C:Ensemble U) (x:U),
In U (Intersection U B C) x -> In U B x /\ In U C x.

Lemma Couple_inv : forall x y z:U, In U (Couple U x y) z -> z = x \/ z = y.

Lemma Setminus_intro :
forall (A B:Ensemble U) (x:U),
In U A x -> ~ In U B x -> In U (Setminus U A B) x.

Lemma Strict_Included_intro :
forall X Y:Ensemble U, Included U X Y /\ X <> Y -> Strict_Included U X Y.

Lemma Strict_Included_strict : forall X:Ensemble U, ~ Strict_Included U X X.

End Ensembles_facts.