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Theorem ssonprc 6626
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 2655 . 2  |-  ( A  e/  _V  <->  -.  A  e.  _V )
2 ssorduni 6620 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
3 ordeleqon 6623 . . . . . . . 8  |-  ( Ord  U. A  <->  ( U. A  e.  On  \/  U. A  =  On ) )
42, 3sylib 196 . . . . . . 7  |-  ( A 
C_  On  ->  ( U. A  e.  On  \/  U. A  =  On ) )
54orcomd 388 . . . . . 6  |-  ( A 
C_  On  ->  ( U. A  =  On  \/  U. A  e.  On ) )
65ord 377 . . . . 5  |-  ( A 
C_  On  ->  ( -. 
U. A  =  On 
->  U. A  e.  On ) )
7 elex 3118 . . . . . 6  |-  ( U. A  e.  On  ->  U. A  e.  _V )
8 uniexb 6609 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
97, 8sylibr 212 . . . . 5  |-  ( U. A  e.  On  ->  A  e.  _V )
106, 9syl6 33 . . . 4  |-  ( A 
C_  On  ->  ( -. 
U. A  =  On 
->  A  e.  _V ) )
1110con1d 124 . . 3  |-  ( A 
C_  On  ->  ( -.  A  e.  _V  ->  U. A  =  On ) )
12 onprc 6619 . . . 4  |-  -.  On  e.  _V
13 uniexg 6596 . . . . 5  |-  ( A  e.  _V  ->  U. A  e.  _V )
14 eleq1 2529 . . . . 5  |-  ( U. A  =  On  ->  ( U. A  e.  _V  <->  On  e.  _V ) )
1513, 14syl5ib 219 . . . 4  |-  ( U. A  =  On  ->  ( A  e.  _V  ->  On  e.  _V ) )
1612, 15mtoi 178 . . 3  |-  ( U. A  =  On  ->  -.  A  e.  _V )
1711, 16impbid1 203 . 2  |-  ( A 
C_  On  ->  ( -.  A  e.  _V  <->  U. A  =  On ) )
181, 17syl5bb 257 1  |-  ( A 
C_  On  ->  ( A  e/  _V  <->  U. A  =  On ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1395    e. wcel 1819    e/ wnel 2653   _Vcvv 3109    C_ wss 3471   U.cuni 4251   Ord word 4886   Oncon0 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891
This theorem is referenced by:  inaprc  9231
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