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Theorem onun2 4182
 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 3513 . 2 ((A On B On) → {A, B} ⊆ On)
2 prexg 3938 . . . 4 ((A On B On) → {A, B} V)
3 ssonuni 4180 . . . 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 3586 . . . 4 ((A On B On) → {A, B} = (AB))
65eleq1d 2103 . . 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 1390  Vcvv 2551   ∪ cun 2909   ⊆ wss 2911  {cpr 3368  ∪ cuni 3571  Oncon0 4066 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071 This theorem is referenced by:  onun2i  4183  rdgon  5913
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