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Theorem prtex 26748
Description: The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem18.1  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
Assertion
Ref Expression
prtex  |-  ( Prt 
A  ->  (  .~  e.  _V  <->  A  e.  _V ) )
Distinct variable group:    x, u, y, A
Allowed substitution hints:    .~ ( x, y, u)

Proof of Theorem prtex
StepHypRef Expression
1 prtlem18.1 . . . 4  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
21prter1 26747 . . 3  |-  ( Prt 
A  ->  .~  Er  U. A )
3 erexb 6685 . . 3  |-  (  .~  Er  U. A  ->  (  .~  e.  _V  <->  U. A  e. 
_V ) )
42, 3syl 15 . 2  |-  ( Prt 
A  ->  (  .~  e.  _V  <->  U. A  e.  _V ) )
5 uniexb 4563 . 2  |-  ( A  e.  _V  <->  U. A  e. 
_V )
64, 5syl6bbr 254 1  |-  ( Prt 
A  ->  (  .~  e.  _V  <->  A  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   U.cuni 3827   {copab 4076    Er wer 6657   Prt wprt 26739
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-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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-er 6660  df-prt 26740
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