Inductive limit_ordinal : Ordinal->Prop :=
| intro_limit_ord: forall {I:Type} (alpha:I->Ordinal),
(forall i:I, exists j:I, alpha i < alpha j) ->
limit_ordinal (ord_sup alpha)
| limit_ord_wd: forall alpha beta:Ordinal,
limit_ordinal alpha -> alpha == beta ->
limit_ordinal beta.

(* Specifications: *)
Lemma ord_successor_or_limit: forall alpha:Ordinal,
successor_ordinal alpha \/ limit_ordinal alpha.
Lemma successor_ordinal_not_limit: forall alpha:Ordinal,
successor_ordinal alpha -> ~ limit_ordinal alpha.