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Theorem ssonuni 6606
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 6605 . 2  |-  ( A 
C_  On  ->  Ord  U. A )
2 uniexg 6581 . . 3  |-  ( A  e.  V  ->  U. A  e.  _V )
3 elong 4886 . . 3  |-  ( U. A  e.  _V  ->  ( U. A  e.  On  <->  Ord  U. A ) )
42, 3syl 16 . 2  |-  ( A  e.  V  ->  ( U. A  e.  On  <->  Ord  U. A ) )
51, 4syl5ibr 221 1  |-  ( A  e.  V  ->  ( A  C_  On  ->  U. A  e.  On ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1767   _Vcvv 3113    C_ wss 3476   U.cuni 4245   Ord word 4877   Oncon0 4878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882
This theorem is referenced by:  ssonunii  6607  onuni  6612  iunon  7009  onfununi  7012  oemapvali  8103  cardprclem  8360  carduni  8362  dfac12lem2  8524  ontgval  29501
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