MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssonuni Unicode version

Theorem ssonuni 4469
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 4468 . 2  |-  ( A 
C_  On  ->  Ord  U. A )
2 uniexg 4408 . . 3  |-  ( A  e.  V  ->  U. A  e.  _V )
3 elong 4293 . . 3  |-  ( U. A  e.  _V  ->  ( U. A  e.  On  <->  Ord  U. A ) )
42, 3syl 17 . 2  |-  ( A  e.  V  ->  ( U. A  e.  On  <->  Ord  U. A ) )
51, 4syl5ibr 214 1  |-  ( A  e.  V  ->  ( A  C_  On  ->  U. A  e.  On ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    e. wcel 1621   _Vcvv 2727    C_ wss 3078   U.cuni 3727   Ord word 4284   Oncon0 4285
This theorem is referenced by:  ssonunii  4470  onuni  4475  iunon  6241  onfununi  6244  oemapvali  7270  cardprclem  7496  carduni  7498  dfac12lem2  7654  ontgval  24044
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289
  Copyright terms: Public domain W3C validator