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Theorem ssonprc 4583
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 2449 . 2  |-  ( A  e/  _V  <->  -.  A  e.  _V )
2 ssorduni 4577 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
3 ordeleqon 4580 . . . . . . . 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 2796 . . . . . 6  |-  ( U. A  e.  On  ->  U. A  e.  _V )
8 uniexb 4563 . . . . . 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 4576 . . . 4  |-  -.  On  e.  _V
13 uniexg 4517 . . . . 5  |-  ( A  e.  _V  ->  U. A  e.  _V )
14 eleq1 2343 . . . . 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 1623    e. wcel 1684    e/ wnel 2447   _Vcvv 2788    C_ wss 3152   U.cuni 3827   Ord word 4391   Oncon0 4392
This theorem is referenced by:  inaprc  8458
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-pow 4188  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-nel 2449  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-pw 3627  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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