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Theorem qsdisj 6981
Description: Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)
Hypotheses
Ref Expression
qsdisj.1  |-  ( ph  ->  R  Er  X )
qsdisj.2  |-  ( ph  ->  B  e.  ( A /. R ) )
qsdisj.3  |-  ( ph  ->  C  e.  ( A /. R ) )
Assertion
Ref Expression
qsdisj  |-  ( ph  ->  ( B  =  C  \/  ( B  i^i  C )  =  (/) ) )

Proof of Theorem qsdisj
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qsdisj.2 . 2  |-  ( ph  ->  B  e.  ( A /. R ) )
2 eqid 2436 . . 3  |-  ( A /. R )  =  ( A /. R
)
3 eqeq1 2442 . . . 4  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =  C  <->  B  =  C
) )
4 ineq1 3535 . . . . 5  |-  ( [ x ] R  =  B  ->  ( [
x ] R  i^i  C )  =  ( B  i^i  C ) )
54eqeq1d 2444 . . . 4  |-  ( [ x ] R  =  B  ->  ( ( [ x ] R  i^i  C )  =  (/)  <->  ( B  i^i  C )  =  (/) ) )
63, 5orbi12d 691 . . 3  |-  ( [ x ] R  =  B  ->  ( ( [ x ] R  =  C  \/  ( [ x ] R  i^i  C )  =  (/) ) 
<->  ( B  =  C  \/  ( B  i^i  C )  =  (/) ) ) )
7 qsdisj.3 . . . . 5  |-  ( ph  ->  C  e.  ( A /. R ) )
87adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  ( A /. R
) )
9 eqeq2 2445 . . . . . 6  |-  ( [ y ] R  =  C  ->  ( [
x ] R  =  [ y ] R  <->  [ x ] R  =  C ) )
10 ineq2 3536 . . . . . . 7  |-  ( [ y ] R  =  C  ->  ( [
x ] R  i^i  [ y ] R )  =  ( [ x ] R  i^i  C ) )
1110eqeq1d 2444 . . . . . 6  |-  ( [ y ] R  =  C  ->  ( ( [ x ] R  i^i  [ y ] R
)  =  (/)  <->  ( [
x ] R  i^i  C )  =  (/) ) )
129, 11orbi12d 691 . . . . 5  |-  ( [ y ] R  =  C  ->  ( ( [ x ] R  =  [ y ] R  \/  ( [ x ] R  i^i  [ y ] R )  =  (/) ) 
<->  ( [ x ] R  =  C  \/  ( [ x ] R  i^i  C )  =  (/) ) ) )
13 qsdisj.1 . . . . . . 7  |-  ( ph  ->  R  Er  X )
1413ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  A )  ->  R  Er  X )
15 erdisj 6952 . . . . . 6  |-  ( R  Er  X  ->  ( [ x ] R  =  [ y ] R  \/  ( [ x ] R  i^i  [ y ] R )  =  (/) ) )
1614, 15syl 16 . . . . 5  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  A )  ->  ( [ x ] R  =  [ y ] R  \/  ( [ x ] R  i^i  [ y ] R )  =  (/) ) )
172, 12, 16ectocld 6971 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  C  e.  ( A /. R
) )  ->  ( [ x ] R  =  C  \/  ( [ x ] R  i^i  C )  =  (/) ) )
188, 17mpdan 650 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( [ x ] R  =  C  \/  ( [ x ] R  i^i  C )  =  (/) ) )
192, 6, 18ectocld 6971 . 2  |-  ( (
ph  /\  B  e.  ( A /. R ) )  ->  ( B  =  C  \/  ( B  i^i  C )  =  (/) ) )
201, 19mpdan 650 1  |-  ( ph  ->  ( B  =  C  \/  ( B  i^i  C )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319   (/)c0 3628    Er wer 6902   [cec 6903   /.cqs 6904
This theorem is referenced by:  qsdisj2  6982  uniinqs  6984  cldsubg  18140  erprt  26722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-er 6905  df-ec 6907  df-qs 6911
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