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Theorem onsseleq 4433
Description: Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
onsseleq  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )

Proof of Theorem onsseleq
StepHypRef Expression
1 eloni 4402 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4402 . 2  |-  ( B  e.  On  ->  Ord  B )
3 ordsseleq 4421 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
41, 2, 3syl2an 463 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   Ord word 4391   Oncon0 4392
This theorem is referenced by:  onsseli  4507  on0eqel  4510  onmindif2  4603  omword  6568  oeword  6588  oewordi  6589  dffi3  7184  cantnflem1d  7390  cantnflem1  7391  r1ord3g  7451  alephdom  7708  cardaleph  7716  cfsmolem  7896  ttukeylem5  8140  alephreg  8204  inar1  8397  gruina  8440  om2uzlt2i  11014
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
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