Definition exists_arbitrarily_large (P: DS_set I -> Prop) :=
forall i:DS_set I, exists j:DS_set I,
DS_ord i j /\ P j.

Lemma not_eventually_eal_not: forall (P: DS_set I -> Prop),
~ eventually P ->
exists_arbitrarily_large (fun i:DS_set I => ~ P i).