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Theorem qsdisj2 7011
Description: A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
qsdisj2  |-  ( R  Er  X  -> Disj  x  e.  ( A /. R
) x )
Distinct variable groups:    x, A    x, X    x, R

Proof of Theorem qsdisj2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpl 445 . . . 4  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  R  Er  X
)
2 simprl 734 . . . 4  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  x  e.  ( A /. R ) )
3 simprr 735 . . . 4  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  y  e.  ( A /. R ) )
41, 2, 3qsdisj 7010 . . 3  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
54ralrimivva 2804 . 2  |-  ( R  Er  X  ->  A. x  e.  ( A /. R
) A. y  e.  ( A /. R
) ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
6 id 21 . . 3  |-  ( x  =  y  ->  x  =  y )
76disjor 4221 . 2  |-  (Disj  x  e.  ( A /. R
) x  <->  A. x  e.  ( A /. R
) A. y  e.  ( A /. R
) ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
85, 7sylibr 205 1  |-  ( R  Er  X  -> Disj  x  e.  ( A /. R
) x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1727   A.wral 2711    i^i cin 3305   (/)c0 3613  Disj wdisj 4207    Er wer 6931   /.cqs 6933
This theorem is referenced by:  qshash  12637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-disj 4208  df-br 4238  df-opab 4292  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-er 6934  df-ec 6936  df-qs 6940
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