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Theorem ssonprc 4599
Description: Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)
Assertion
Ref Expression
ssonprc  |-  ( A 
C_  On  ->  ( A  e/  _V  <->  U. A  =  On ) )

Proof of Theorem ssonprc
StepHypRef Expression
1 df-nel 2462 . 2  |-  ( A  e/  _V  <->  -.  A  e.  _V )
2 ssorduni 4593 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
3 ordeleqon 4596 . . . . . . . 8  |-  ( Ord  U. A  <->  ( U. A  e.  On  \/  U. A  =  On ) )
42, 3sylib 188 . . . . . . 7  |-  ( A 
C_  On  ->  ( U. A  e.  On  \/  U. A  =  On ) )
54orcomd 377 . . . . . 6  |-  ( A 
C_  On  ->  ( U. A  =  On  \/  U. A  e.  On ) )
65ord 366 . . . . 5  |-  ( A 
C_  On  ->  ( -. 
U. A  =  On 
->  U. A  e.  On ) )
7 elex 2809 . . . . . 6  |-  ( U. A  e.  On  ->  U. A  e.  _V )
8 uniexb 4579 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
97, 8sylibr 203 . . . . 5  |-  ( U. A  e.  On  ->  A  e.  _V )
106, 9syl6 29 . . . 4  |-  ( A 
C_  On  ->  ( -. 
U. A  =  On 
->  A  e.  _V ) )
1110con1d 116 . . 3  |-  ( A 
C_  On  ->  ( -.  A  e.  _V  ->  U. A  =  On ) )
12 onprc 4592 . . . 4  |-  -.  On  e.  _V
13 uniexg 4533 . . . . 5  |-  ( A  e.  _V  ->  U. A  e.  _V )
14 eleq1 2356 . . . . 5  |-  ( U. A  =  On  ->  ( U. A  e.  _V  <->  On  e.  _V ) )
1513, 14syl5ib 210 . . . 4  |-  ( U. A  =  On  ->  ( A  e.  _V  ->  On  e.  _V ) )
1612, 15mtoi 169 . . 3  |-  ( U. A  =  On  ->  -.  A  e.  _V )
1711, 16impbid1 194 . 2  |-  ( A 
C_  On  ->  ( -.  A  e.  _V  <->  U. A  =  On ) )
181, 17syl5bb 248 1  |-  ( A 
C_  On  ->  ( A  e/  _V  <->  U. A  =  On ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1632    e. wcel 1696    e/ wnel 2460   _Vcvv 2801    C_ wss 3165   U.cuni 3843   Ord word 4407   Oncon0 4408
This theorem is referenced by:  inaprc  8474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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