Theorem ordelsuc 7020

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 7018	. . 3 |- ( Ord B -> ( A e. B -> suc A C_ B ) )
2	1	adantl 482	. 2 |- ( ( A e. C /\ Ord B ) -> ( A e. B -> suc A C_ B ) )
3		sucssel 5819	. . 3 |- ( A e. C -> ( suc A C_ B -> A e. B ) )
4	3	adantr 481	. 2 |- ( ( A e. C /\ Ord B ) -> ( suc A C_ B -> A e. B ) )
5	2, 4	impbid 202	1 |- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   Ord word 5722   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  onsucmin  7021  onsucssi  7041  tfindsg2  7061  ordgt0ge1  7577  onomeneq  8150  cantnflem1  8586  r1ordg  8641  r1val1  8649  rankonidlem  8691  rankxplim3  8744