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Theorem onun2 4164
Description: The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
Assertion
Ref Expression
onun2 ((A On B On) → (AB) On)

Proof of Theorem onun2
StepHypRef Expression
1 prssi 3495 . 2 ((A On B On) → {A, B} ⊆ On)
2 prexg 3920 . . . 4 ((A On B On) → {A, B} V)
3 ssonuni 4162 . . . 4 ({A, B} V → ({A, B} ⊆ On → {A, B} On))
42, 3syl 14 . . 3 ((A On B On) → ({A, B} ⊆ On → {A, B} On))
5 uniprg 3568 . . . 4 ((A On B On) → {A, B} = (AB))
65eleq1d 2089 . . 3 ((A On B On) → ( {A, B} On ↔ (AB) On))
74, 6sylibd 138 . 2 ((A On B On) → ({A, B} ⊆ On → (AB) On))
81, 7mpd 13 1 ((A On B On) → (AB) On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1375  Vcvv 2534  cun 2891  wss 2893  {cpr 3350   cuni 3553  Oncon0 4047
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 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pr 3917  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-sn 3355  df-pr 3356  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052
This theorem is referenced by:  onun2i  4165  rdgon  5888
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