From Coq Require Export Relation_Definitions QArith ZArith.
ProofIrrelevance ClassicalChoice.
DependentTypeChoice.
From ZornsLemma Require Export FiniteTypes IndexedFamilies.
Local Close Scope Q_scope.
Set Asymmetric Patterns.
Inductive CountableT (X : Type) : Prop :=
| intro_nat_injection (f : X -> nat) : injective f -> CountableT X.
Definition Countable {X : Type} (S : Ensemble X) : Prop :=
CountableT {x:X | In S x}.

Lemma countable_union: forall {X A:Type}
(F:IndexedFamily A X), CountableT A ->
(forall a:A, Countable (F a)) ->
Countable (IndexedUnion F).