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Theorem qsdisj2 6949
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 444 . . . 4  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  R  Er  X
)
2 simprl 733 . . . 4  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  x  e.  ( A /. R ) )
3 simprr 734 . . . 4  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  y  e.  ( A /. R ) )
41, 2, 3qsdisj 6948 . . 3  |-  ( ( R  Er  X  /\  ( x  e.  ( A /. R )  /\  y  e.  ( A /. R ) ) )  ->  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
54ralrimivva 2766 . 2  |-  ( R  Er  X  ->  A. x  e.  ( A /. R
) A. y  e.  ( A /. R
) ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )
6 id 20 . . 3  |-  ( x  =  y  ->  x  =  y )
76disjor 4164 . 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 204 1  |-  ( R  Er  X  -> Disj  x  e.  ( A /. R
) x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674    i^i cin 3287   (/)c0 3596  Disj wdisj 4150    Er wer 6869   /.cqs 6871
This theorem is referenced by:  qshash  12569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-disj 4151  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-er 6872  df-ec 6874  df-qs 6878
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