Theorem ordelsuc 6673

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 6671	. . 3 |- ( Ord B -> ( A e. B -> suc A C_ B ) )
2	1	adantl 472	. 2 |- ( ( A e. C /\ Ord B ) -> ( A e. B -> suc A C_ B ) )
3		sucssel 5533	. . 3 |- ( A e. C -> ( suc A C_ B -> A e. B ) )
4	3	adantr 471	. 2 |- ( ( A e. C /\ Ord B ) -> ( suc A C_ B -> A e. B ) )
5	2, 4	impbid 195	1 |- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   e. wcel 1897   C_ wss 3415   Ord word 5440   suc csuc 5443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-tr 4511  df-eprel 4763  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-ord 5444  df-on 5445  df-suc 5447

This theorem is referenced by:  onsucmin  6674  onsucssi  6694  tfindsg2  6714  ordgt0ge1  7224  onomeneq  7787  cantnflem1  8219  r1ordg  8274  r1val1  8282  rankonidlem  8324  rankxplim3  8377