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Theorem omelon 9106
Description: Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
Assertion
Ref Expression
omelon ω ∈ On

Proof of Theorem omelon
StepHypRef Expression
1 omex 9103 . 2 ω ∈ V
2 omelon2 7586 . 2 (ω ∈ V → ω ∈ On)
31, 2ax-mp 5 1 ω ∈ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  Vcvv 3480  Oncon0 6178  ωcom 7574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455  ax-inf2 9101
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-om 7575
This theorem is referenced by:  oancom  9111  cnfcomlem  9159  cnfcom  9160  cnfcom2lem  9161  cnfcom2  9162  cnfcom3lem  9163  cnfcom3  9164  cnfcom3clem  9165  cardom  9412  infxpenlem  9437  xpomen  9439  infxpidm2  9441  infxpenc  9442  infxpenc2lem1  9443  infxpenc2  9446  alephon  9493  infenaleph  9515  iunfictbso  9538  dfac12k  9571  infunsdom1  9633  domtriomlem  9862  iunctb  9994  pwcfsdom  10003  canthp1lem2  10073  pwfseqlem4a  10081  pwfseqlem4  10082  pwfseqlem5  10083  wunex3  10161  znnen  15565  qnnen  15566  cygctb  19012  2ndcctbss  22063  2ndcomap  22066  2ndcsep  22067  tx1stc  22258  tx2ndc  22259  met1stc  23131  met2ndci  23132  re2ndc  23409  uniiccdif  24185  dyadmbl  24207  opnmblALT  24210  mbfimaopnlem  24262  aannenlem3  24929  exrecfnlem  34741  poimirlem32  35034  numinfctb  39963  infordmin  40156  aleph1min  40172  alephiso3  40174
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