Require Export Ensembles.
Set Implicit Arguments.
Section Families.
Variable T:Type.
Definition Family := Ensemble (Ensemble T).
Variable F:Family.
Inductive FamilyUnion: Ensemble T :=
| family_union_intro: forall (S:Ensemble T) (x:T),
In F S -> In S x -> In FamilyUnion x.
Inductive FamilyIntersection: Ensemble T :=
| family_intersection_intro: forall x:T,
(forall S:Ensemble T, In F S -> In S x) ->
In FamilyIntersection x.
End Families.
Section FamilyFacts.
Variable T:Type.

Lemma empty_family_intersection:
FamilyIntersection (@Empty_set (Ensemble T)) = Full_set.