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Theorem ssonuni 4753
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 4752 . 2  |-  ( A 
C_  On  ->  Ord  U. A )
2 uniexg 4692 . . 3  |-  ( A  e.  V  ->  U. A  e.  _V )
3 elong 4576 . . 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 213 1  |-  ( A  e.  V  ->  ( A  C_  On  ->  U. A  e.  On ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725   _Vcvv 2943    C_ wss 3307   U.cuni 4002   Ord word 4567   Oncon0 4568
This theorem is referenced by:  ssonunii  4754  onuni  4759  iunon  6586  onfununi  6589  oemapvali  7624  cardprclem  7850  carduni  7852  dfac12lem2  8008  ontgval  26124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-tr 4290  df-eprel 4481  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572
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