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Theorem onun2 4307
Description: The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
Assertion
Ref Expression
onun2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  u.  B
)  e.  On )

Proof of Theorem onun2
StepHypRef Expression
1 prssi 3595 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  { A ,  B }  C_  On )
2 prexg 4038 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  { A ,  B }  e.  _V )
3 ssonuni 4305 . . . 4  |-  ( { A ,  B }  e.  _V  ->  ( { A ,  B }  C_  On  ->  U. { A ,  B }  e.  On ) )
42, 3syl 14 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( { A ,  B }  C_  On  ->  U. { A ,  B }  e.  On )
)
5 uniprg 3668 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  U. { A ,  B }  =  ( A  u.  B )
)
65eleq1d 2156 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( U. { A ,  B }  e.  On  <->  ( A  u.  B )  e.  On ) )
74, 6sylibd 147 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( { A ,  B }  C_  On  ->  ( A  u.  B )  e.  On ) )
81, 7mpd 13 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  u.  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   _Vcvv 2619    u. cun 2997    C_ wss 2999   {cpr 3447   U.cuni 3653   Oncon0 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195
This theorem is referenced by:  onun2i  4308  rdgon  6151
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