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Theorem ordelsuc 4791
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
StepHypRef Expression
1 ordsucss 4789 . . 3  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
21adantl 453 . 2  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
3 sucssel 4665 . . 3  |-  ( A  e.  C  ->  ( suc  A  C_  B  ->  A  e.  B ) )
43adantr 452 . 2  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( suc  A  C_  B  ->  A  e.  B ) )
52, 4impbid 184 1  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    C_ wss 3312   Ord word 4572   suc csuc 4575
This theorem is referenced by:  onsucmin  4792  onsucssi  4812  tfindsg2  4832  ordgt0ge1  6732  onomeneq  7287  omsucdomOLD  7293  cantnflem1  7634  r1ordg  7693  r1val1  7701  rankonidlem  7743  rankxplim3  7794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
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