Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  omelon Structured version   Visualization version   GIF version

Theorem omelon 8581
 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 8578 . 2 ω ∈ V
2 omelon2 7119 . 2 (ω ∈ V → ω ∈ On)
31, 2ax-mp 5 1 ω ∈ On
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2030  Vcvv 3231  Oncon0 5761  ωcom 7107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-inf2 8576 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-om 7108 This theorem is referenced by:  oancom  8586  cnfcomlem  8634  cnfcom  8635  cnfcom2lem  8636  cnfcom2  8637  cnfcom3lem  8638  cnfcom3  8639  cnfcom3clem  8640  cardom  8850  infxpenlem  8874  xpomen  8876  infxpidm2  8878  infxpenc  8879  infxpenc2lem1  8880  infxpenc2  8883  alephon  8930  infenaleph  8952  iunfictbso  8975  dfac12k  9007  infunsdom1  9073  domtriomlem  9302  iunctb  9434  pwcfsdom  9443  canthp1lem2  9513  pwfseqlem4a  9521  pwfseqlem4  9522  pwfseqlem5  9523  wunex3  9601  znnen  14985  qnnen  14986  cygctb  18339  2ndcctbss  21306  2ndcomap  21309  2ndcsep  21310  tx1stc  21501  tx2ndc  21502  met1stc  22373  met2ndci  22374  re2ndc  22651  uniiccdif  23392  dyadmbl  23414  opnmblALT  23417  mbfimaopnlem  23467  aannenlem3  24130  poimirlem32  33571  numinfctb  37990
 Copyright terms: Public domain W3C validator