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Theorem prtex 26851
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 26850 . . 3  |-  ( Prt 
A  ->  .~  Er  U. A )
3 erexb 6701 . . 3  |-  (  .~  Er  U. A  ->  (  .~  e.  _V  <->  U. A  e. 
_V ) )
42, 3syl 15 . 2  |-  ( Prt 
A  ->  (  .~  e.  _V  <->  U. A  e.  _V ) )
5 uniexb 4579 . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   U.cuni 3843   {copab 4092    Er wer 6673   Prt wprt 26842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-er 6676  df-prt 26843
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