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Theorem onsseli 4507
Description: Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)
Hypotheses
Ref Expression
on.1  |-  A  e.  On
on.2  |-  B  e.  On
Assertion
Ref Expression
onsseli  |-  ( A 
C_  B  <->  ( A  e.  B  \/  A  =  B ) )

Proof of Theorem onsseli
StepHypRef Expression
1 on.1 . 2  |-  A  e.  On
2 on.2 . 2  |-  B  e.  On
3 onsseleq 4433 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
41, 2, 3mp2an 653 1  |-  ( A 
C_  B  <->  ( A  e.  B  \/  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684    C_ wss 3152   Oncon0 4392
This theorem is referenced by:  cardom  7619  tskcard  8403
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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