Definition IndexedFamily := A -> Ensemble T.Inductive IndexedUnion : Ensemble T :=
  | indexed_union_intro: forall (a:A) (x:T),
    In (F a) x -> In IndexedUnion x.Inductive IndexedUnion : Ensemble T :=
  | indexed_union_intro: forall (a:A) (x:T),
    In (F a) x -> In IndexedUnion x.Inductive IndexedIntersection : Ensemble T :=
  | indexed_intersection_intro: forall (x:T),
    (forall a:A, In (F a) x) -> In IndexedIntersection x.Inductive IndexedIntersection : Ensemble T :=
  | indexed_intersection_intro: forall (x:T),
    (forall a:A, In (F a) x) -> In IndexedIntersection x.

Lemma empty_indexed_union: forall {T:Type} (F:IndexedFamily False T), IndexedUnion F = Empty_set.