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Theorem onun2 4166
Description: The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
Assertion
Ref Expression
onun2  On  On  u.  On

Proof of Theorem onun2
StepHypRef Expression
1 prssi 3496 . 2  On  On  { ,  }  C_  On
2 prexg 3921 . . . 4  On  On  { ,  }  _V
3 ssonuni 4164 . . . 4  { ,  }  _V  { ,  }  C_  On  U. { ,  }  On
42, 3syl 14 . . 3  On  On  { ,  }  C_  On  U. { ,  }  On
5 uniprg 3569 . . . 4  On  On  U. { ,  }  u.
65eleq1d 2088 . . 3  On  On  U. { ,  }  On  u.  On
74, 6sylibd 138 . 2  On  On  { ,  }  C_  On  u.  On
81, 7mpd 13 1  On  On  u.  On
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wcel 1374   _Vcvv 2535    u. cun 2892    C_ wss 2894   {cpr 3351   U.cuni 3554   Oncon0 4049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054
This theorem is referenced by:  onun2i  4167  rdgon  5893
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