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Theorem ssonprc 6646
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 2636 . 2  |-  ( A  e/  _V  <->  -.  A  e.  _V )
2 ssorduni 6639 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
3 ordeleqon 6642 . . . . . . . 8  |-  ( Ord  U. A  <->  ( U. A  e.  On  \/  U. A  =  On ) )
42, 3sylib 201 . . . . . . 7  |-  ( A 
C_  On  ->  ( U. A  e.  On  \/  U. A  =  On ) )
54orcomd 394 . . . . . 6  |-  ( A 
C_  On  ->  ( U. A  =  On  \/  U. A  e.  On ) )
65ord 383 . . . . 5  |-  ( A 
C_  On  ->  ( -. 
U. A  =  On 
->  U. A  e.  On ) )
7 elex 3066 . . . . . 6  |-  ( U. A  e.  On  ->  U. A  e.  _V )
8 uniexb 6628 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
97, 8sylibr 217 . . . . 5  |-  ( U. A  e.  On  ->  A  e.  _V )
106, 9syl6 34 . . . 4  |-  ( A 
C_  On  ->  ( -. 
U. A  =  On 
->  A  e.  _V ) )
1110con1d 129 . . 3  |-  ( A 
C_  On  ->  ( -.  A  e.  _V  ->  U. A  =  On ) )
12 onprc 6638 . . . 4  |-  -.  On  e.  _V
13 uniexg 6615 . . . . 5  |-  ( A  e.  _V  ->  U. A  e.  _V )
14 eleq1 2528 . . . . 5  |-  ( U. A  =  On  ->  ( U. A  e.  _V  <->  On  e.  _V ) )
1513, 14syl5ib 227 . . . 4  |-  ( U. A  =  On  ->  ( A  e.  _V  ->  On  e.  _V ) )
1612, 15mtoi 183 . . 3  |-  ( U. A  =  On  ->  -.  A  e.  _V )
1711, 16impbid1 208 . 2  |-  ( A 
C_  On  ->  ( -.  A  e.  _V  <->  U. A  =  On ) )
181, 17syl5bb 265 1  |-  ( A 
C_  On  ->  ( A  e/  _V  <->  U. A  =  On ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    = wceq 1455    e. wcel 1898    e/ wnel 2634   _Vcvv 3057    C_ wss 3416   U.cuni 4212   Ord word 5441   Oncon0 5442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-tr 4512  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446
This theorem is referenced by:  inaprc  9287
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