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Theorem prtex 30452
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 30451 . . 3  |-  ( Prt 
A  ->  .~  Er  U. A )
3 erexb 7337 . . 3  |-  (  .~  Er  U. A  ->  (  .~  e.  _V  <->  U. A  e. 
_V ) )
42, 3syl 16 . 2  |-  ( Prt 
A  ->  (  .~  e.  _V  <->  U. A  e.  _V ) )
5 uniexb 6595 . 2  |-  ( A  e.  _V  <->  U. A  e. 
_V )
64, 5syl6bbr 263 1  |-  ( Prt 
A  ->  (  .~  e.  _V  <->  A  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   U.cuni 4245   {copab 4504    Er wer 7309   Prt wprt 30443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-er 7312  df-prt 30444
This theorem is referenced by: (None)
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