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Theorem erprt 26724
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 445 . . . 4  |-  ( (  .~  Er  X  /\  ( x  e.  ( A /.  .~  )  /\  y  e.  ( A /.  .~  ) ) )  ->  .~  Er  X
)
2 simprl 734 . . . 4  |-  ( (  .~  Er  X  /\  ( x  e.  ( A /.  .~  )  /\  y  e.  ( A /.  .~  ) ) )  ->  x  e.  ( A /.  .~  )
)
3 simprr 735 . . . 4  |-  ( (  .~  Er  X  /\  ( x  e.  ( A /.  .~  )  /\  y  e.  ( A /.  .~  ) ) )  ->  y  e.  ( A /.  .~  )
)
41, 2, 3qsdisj 6983 . . 3  |-  ( (  .~  Er  X  /\  ( x  e.  ( A /.  .~  )  /\  y  e.  ( A /.  .~  ) ) )  ->  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
54ralrimivva 2800 . 2  |-  (  .~  Er  X  ->  A. x  e.  ( A /.  .~  ) A. y  e.  ( A /.  .~  )
( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
6 df-prt 26723 . 2  |-  ( Prt  ( A /.  .~  ) 
<-> 
A. x  e.  ( A /.  .~  ) A. y  e.  ( A /.  .~  ) ( x  =  y  \/  ( x  i^i  y
)  =  (/) ) )
75, 6sylibr 205 1  |-  (  .~  Er  X  ->  Prt  ( A /.  .~  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321   (/)c0 3630    Er wer 6904   /.cqs 6906   Prt wprt 26722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-er 6907  df-ec 6909  df-qs 6913  df-prt 26723
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