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Theorem erprt 26844
Description: The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
erprt  |-  (  .~  Er  X  ->  Prt  ( A /.  .~  ) )

Proof of Theorem erprt
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . . 4  |-  ( (  .~  Er  X  /\  ( x  e.  ( A /.  .~  )  /\  y  e.  ( A /.  .~  ) ) )  ->  .~  Er  X
)
2 simprl 732 . . . 4  |-  ( (  .~  Er  X  /\  ( x  e.  ( A /.  .~  )  /\  y  e.  ( A /.  .~  ) ) )  ->  x  e.  ( A /.  .~  )
)
3 simprr 733 . . . 4  |-  ( (  .~  Er  X  /\  ( x  e.  ( A /.  .~  )  /\  y  e.  ( A /.  .~  ) ) )  ->  y  e.  ( A /.  .~  )
)
41, 2, 3qsdisj 6752 . . 3  |-  ( (  .~  Er  X  /\  ( x  e.  ( A /.  .~  )  /\  y  e.  ( A /.  .~  ) ) )  ->  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
54ralrimivva 2648 . 2  |-  (  .~  Er  X  ->  A. x  e.  ( A /.  .~  ) A. y  e.  ( A /.  .~  )
( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
6 df-prt 26843 . 2  |-  ( Prt  ( A /.  .~  ) 
<-> 
A. x  e.  ( A /.  .~  ) A. y  e.  ( A /.  .~  ) ( x  =  y  \/  ( x  i^i  y
)  =  (/) ) )
75, 6sylibr 203 1  |-  (  .~  Er  X  ->  Prt  ( A /.  .~  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164   (/)c0 3468    Er wer 6673   /.cqs 6675   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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  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-res 4717  df-ima 4718  df-er 6676  df-ec 6678  df-qs 6682  df-prt 26843
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