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Theorem erprt 26741
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 6736 . . 3  |-  ( (  .~  Er  X  /\  ( x  e.  ( A /.  .~  )  /\  y  e.  ( A /.  .~  ) ) )  ->  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
54ralrimivva 2635 . 2  |-  (  .~  Er  X  ->  A. x  e.  ( A /.  .~  ) A. y  e.  ( A /.  .~  )
( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
6 df-prt 26740 . 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 1623    e. wcel 1684   A.wral 2543    i^i cin 3151   (/)c0 3455    Er wer 6657   /.cqs 6659   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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  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-res 4701  df-ima 4702  df-er 6660  df-ec 6662  df-qs 6666  df-prt 26740
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