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Theorem ssonprc 4772
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 2602 . 2  |-  ( A  e/  _V  <->  -.  A  e.  _V )
2 ssorduni 4766 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
3 ordeleqon 4769 . . . . . . . 8  |-  ( Ord  U. A  <->  ( U. A  e.  On  \/  U. A  =  On ) )
42, 3sylib 189 . . . . . . 7  |-  ( A 
C_  On  ->  ( U. A  e.  On  \/  U. A  =  On ) )
54orcomd 378 . . . . . 6  |-  ( A 
C_  On  ->  ( U. A  =  On  \/  U. A  e.  On ) )
65ord 367 . . . . 5  |-  ( A 
C_  On  ->  ( -. 
U. A  =  On 
->  U. A  e.  On ) )
7 elex 2964 . . . . . 6  |-  ( U. A  e.  On  ->  U. A  e.  _V )
8 uniexb 4752 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
97, 8sylibr 204 . . . . 5  |-  ( U. A  e.  On  ->  A  e.  _V )
106, 9syl6 31 . . . 4  |-  ( A 
C_  On  ->  ( -. 
U. A  =  On 
->  A  e.  _V ) )
1110con1d 118 . . 3  |-  ( A 
C_  On  ->  ( -.  A  e.  _V  ->  U. A  =  On ) )
12 onprc 4765 . . . 4  |-  -.  On  e.  _V
13 uniexg 4706 . . . . 5  |-  ( A  e.  _V  ->  U. A  e.  _V )
14 eleq1 2496 . . . . 5  |-  ( U. A  =  On  ->  ( U. A  e.  _V  <->  On  e.  _V ) )
1513, 14syl5ib 211 . . . 4  |-  ( U. A  =  On  ->  ( A  e.  _V  ->  On  e.  _V ) )
1612, 15mtoi 171 . . 3  |-  ( U. A  =  On  ->  -.  A  e.  _V )
1711, 16impbid1 195 . 2  |-  ( A 
C_  On  ->  ( -.  A  e.  _V  <->  U. A  =  On ) )
181, 17syl5bb 249 1  |-  ( A 
C_  On  ->  ( A  e/  _V  <->  U. A  =  On ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1652    e. wcel 1725    e/ wnel 2600   _Vcvv 2956    C_ wss 3320   U.cuni 4015   Ord word 4580   Oncon0 4581
This theorem is referenced by:  inaprc  8711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
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