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Theorem qsdisj2 6739
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 443 . . . 4  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  R  Er  X
)
2 simprl 732 . . . 4  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  x  e.  ( A /. R ) )
3 simprr 733 . . . 4  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  y  e.  ( A /. R ) )
41, 2, 3qsdisj 6738 . . 3  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
54ralrimivva 2637 . 2  |-  ( R  Er  X  ->  A. x  e.  ( A /. R
) A. y  e.  ( A /. R
) ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
6 id 19 . . 3  |-  ( x  =  y  ->  x  =  y )
76disjor 4009 . 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 203 1  |-  ( R  Er  X  -> Disj  x  e.  ( A /. R
) x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1625    e. wcel 1686   A.wral 2545    i^i cin 3153   (/)c0 3457  Disj wdisj 3995    Er wer 6659   /.cqs 6661
This theorem is referenced by:  qshash  12287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-disj 3996  df-br 4026  df-opab 4080  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-er 6662  df-ec 6664  df-qs 6668
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