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Theorem onsssuc 4480
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 4402 . 2  |-  ( B  e.  On  ->  Ord  B )
2 ordsssuc 4479 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
31, 2sylan2 460 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    C_ wss 3152   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  ordsssuc2  4481  onmindif  4482  tfindsg  4651  dfom2  4658  findsg  4683  ondif2  6501  oeeui  6600  cantnflem1  7391  rankr1bg  7475  rankr1c  7493  cofsmo  7895  cfsmolem  7896  cfcof  7900  fin1a2lem9  8034  alephreg  8204  winainflem  8315  nobndlem8  24353  onsuct0  24880  onint1  24888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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