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Theorem onun2 4154
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 3485 . 2 ((A On B On) → {A, B} ⊆ On)
2 prexg 3910 . . . 4 ((A On B On) → {A, B} V)
3 ssonuni 4152 . . . 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 3558 . . . 4 ((A On B On) → {A, B} = (AB))
65eleq1d 2079 . . 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 1366  Vcvv 2526  cun 2883  wss 2885  {cpr 3340   cuni 3543  Oncon0 4038
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pr 3907  ax-un 4108
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-sn 3345  df-pr 3346  df-uni 3544  df-tr 3818  df-iord 4041  df-on 4043
This theorem is referenced by:  onun2i  4155  rdgon  5881
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