Theorem ordelsuc 4429

Description: A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)

Assertion

Ref	Expression
ordelsuc	|- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )

Proof of Theorem ordelsuc

Step	Hyp	Ref	Expression
1		ordsucss 4428	. . 3 |- ( Ord B -> ( A e. B -> suc A C_ B ) )
2	1	adantl 275	. 2 |- ( ( A e. C /\ Ord B ) -> ( A e. B -> suc A C_ B ) )
3		sucssel 4354	. . 3 |- ( A e. C -> ( suc A C_ B -> A e. B ) )
4	3	adantr 274	. 2 |- ( ( A e. C /\ Ord B ) -> ( suc A C_ B -> A e. B ) )
5	2, 4	impbid 128	1 |- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1481   C_ wss 3076   Ord word 4292   suc csuc 4295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-uni 3745  df-tr 4035  df-iord 4296  df-suc 4301

This theorem is referenced by:  onsucssi  4430  onsucmin  4431  onsucelsucr  4432  onsucsssucr  4433  onsucsssucexmid  4450  frecsuclem  6311  ordgt0ge1  6340  nnsucsssuc  6396  ennnfonelemk  11949  nninfsellemeq  13385