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Theorem qsdisj2 6879
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 6878 . . 3  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
54ralrimivva 2720 . 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 4109 . 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 1647    e. wcel 1715   A.wral 2628    i^i cin 3237   (/)c0 3543  Disj wdisj 4095    Er wer 6799   /.cqs 6801
This theorem is referenced by:  qshash  12493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-disj 4096  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-er 6802  df-ec 6804  df-qs 6808
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