Inductive Cardinal : Type :=
  | cardinality: Type -> Cardinal.Inductive Cardinal : Type :=
  | cardinality: Type -> Cardinal.Fixpoint n_element_set (n:nat) : Set :=
  match n with
  | O => False
  | S m => option (n_element_set m)
  end.Definition nat_to_cardinal (n:nat) :=
  cardinality (n_element_set n).Definition aleph0 := cardinality nat.Inductive eq_cardinal : Cardinal -> Cardinal -> Prop :=
  | bij_eq_cardinal: forall {X Y:Type} (f:X->Y),
    bijective f -> eq_cardinal (cardinality X) (cardinality Y).Inductive eq_cardinal : Cardinal -> Cardinal -> Prop :=
  | bij_eq_cardinal: forall {X Y:Type} (f:X->Y),
    bijective f -> eq_cardinal (cardinality X) (cardinality Y).Inductive le_cardinal : Cardinal -> Cardinal -> Prop :=
  | inj_le_cardinal: forall {X Y:Type} (f:X->Y),
    injective f -> le_cardinal (cardinality X) (cardinality Y).Inductive le_cardinal : Cardinal -> Cardinal -> Prop :=
  | inj_le_cardinal: forall {X Y:Type} (f:X->Y),
    injective f -> le_cardinal (cardinality X) (cardinality Y).Definition lt_cardinal (kappa lambda:Cardinal) : Prop :=
  le_cardinal kappa lambda /\ ~ eq_cardinal kappa lambda.Definition ge_cardinal (kappa lambda:Cardinal) : Prop :=
  le_cardinal lambda kappa.Definition gt_cardinal (kappa lambda:Cardinal) : Prop :=
  lt_cardinal lambda kappa.

Lemma surj_le_cardinal: forall {X Y:Type} (f:X->Y), surjective f -> le_cardinal (cardinality Y) (cardinality X).