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Theorem ssonuni 4578
Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
Assertion
Ref Expression
ssonuni  |-  ( A  e.  V  ->  ( A  C_  On  ->  U. A  e.  On ) )

Proof of Theorem ssonuni
StepHypRef Expression
1 ssorduni 4577 . 2  |-  ( A 
C_  On  ->  Ord  U. A )
2 uniexg 4517 . . 3  |-  ( A  e.  V  ->  U. A  e.  _V )
3 elong 4400 . . 3  |-  ( U. A  e.  _V  ->  ( U. A  e.  On  <->  Ord  U. A ) )
42, 3syl 15 . 2  |-  ( A  e.  V  ->  ( U. A  e.  On  <->  Ord  U. A ) )
51, 4syl5ibr 212 1  |-  ( A  e.  V  ->  ( A  C_  On  ->  U. A  e.  On ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   _Vcvv 2788    C_ wss 3152   U.cuni 3827   Ord word 4391   Oncon0 4392
This theorem is referenced by:  ssonunii  4579  onuni  4584  iunon  6355  onfununi  6358  oemapvali  7386  cardprclem  7612  carduni  7614  dfac12lem2  7770  ontgval  24870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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