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Theorem onsssuc 4954
Description: A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
onsssuc  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )

Proof of Theorem onsssuc
StepHypRef Expression
1 eloni 4877 . 2  |-  ( B  e.  On  ->  Ord  B )
2 ordsssuc 4953 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
31, 2sylan2 472 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823    C_ wss 3461   Ord word 4866   Oncon0 4867   suc csuc 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873
This theorem is referenced by:  ordsssuc2  4955  onmindif  4956  tfindsg  6668  dfom2  6675  findsg  6700  ondif2  7144  oeeui  7243  cantnflem1  8099  cantnflem1OLD  8122  rankr1bg  8212  rankr1c  8230  cofsmo  8640  cfsmolem  8641  cfcof  8645  fin1a2lem9  8779  alephreg  8948  winainflem  9060  nobndlem8  29699  onsuct0  30134  onint1  30142
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