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Theorem prtex 32204
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 32203 . . 3  |-  ( Prt 
A  ->  .~  Er  U. A )
3 erexb 7387 . . 3  |-  (  .~  Er  U. A  ->  (  .~  e.  _V  <->  U. A  e. 
_V ) )
42, 3syl 17 . 2  |-  ( Prt 
A  ->  (  .~  e.  _V  <->  U. A  e.  _V ) )
5 uniexb 6606 . 2  |-  ( A  e.  _V  <->  U. A  e. 
_V )
64, 5syl6bbr 266 1  |-  ( Prt 
A  ->  (  .~  e.  _V  <->  A  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   E.wrex 2774   _Vcvv 3078   U.cuni 4213   {copab 4474    Er wer 7359   Prt wprt 32195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-er 7362  df-prt 32196
This theorem is referenced by: (None)
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