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Theorem onun2 4139
 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 3473 . 2 ((A On B On) → {A, B} ⊆ On)
2 prexg 3899 . . . 4 ((A On B On) → {A, B} V)
3 ssonuni 4137 . . . 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 3547 . . . 4 ((A On B On) → {A, B} = (AB))
65eleq1d 2088 . . 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 2533   ∪ cun 2893   ⊆ wss 2895  {cpr 3328  ∪ cuni 3532  Oncon0 4024 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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pr 3896  ax-un 4093 This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-sn 3333  df-pr 3334  df-uni 3533  df-tr 3807  df-iord 4027  df-on 4028 This theorem is referenced by:  onun2i  4140  rdgon  5859
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