proposition: Lemma cnextE n: ordinalp n -> cnext (\aleph n) = \aleph (osucc n).
proof: Proof. have ilx:= iclosed_non_coll_infinite_c. move/ (iclosed_col_fs ilx) => [xy icx icy h]. move: (icx)(icy) => [cx _] [cy _]. move: (cnext_pr1 cx)=> [ua ub uc]; apply: cleA. apply: uc; apply:(oclt3 cx cy xy). have sa:=(cle_inf_inf icx (proj1 ub)). exact:(ocle3 cy ua (h _ sa (oclt ub))). Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat)