Definition injective (f:U -> V) := forall x y:U, f x = f y -> x = y.

Lemma Pigeonhole_principle : forall (A:Ensemble U) (f:U -> V) (n:nat), cardinal _ A n -> forall n':nat, cardinal _ (Im A f) n' -> n' < n -> exists x : _, (exists y : _, f x = f y /\ x <> y).