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Theorem qsdisj 6736
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 2283 . . 3  |-  ( A /. R )  =  ( A /. R
)
3 eqeq1 2289 . . . 4  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =  C  <->  B  =  C
) )
4 ineq1 3363 . . . . 5  |-  ( [ x ] R  =  B  ->  ( [
x ] R  i^i  C )  =  ( B  i^i  C ) )
54eqeq1d 2291 . . . 4  |-  ( [ x ] R  =  B  ->  ( ( [ x ] R  i^i  C )  =  (/)  <->  ( B  i^i  C )  =  (/) ) )
63, 5orbi12d 690 . . 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 451 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  ( A /. R
) )
9 eqeq2 2292 . . . . . 6  |-  ( [ y ] R  =  C  ->  ( [
x ] R  =  [ y ] R  <->  [ x ] R  =  C ) )
10 ineq2 3364 . . . . . . 7  |-  ( [ y ] R  =  C  ->  ( [
x ] R  i^i  [ y ] R )  =  ( [ x ] R  i^i  C ) )
1110eqeq1d 2291 . . . . . 6  |-  ( [ y ] R  =  C  ->  ( ( [ x ] R  i^i  [ y ] R
)  =  (/)  <->  ( [
x ] R  i^i  C )  =  (/) ) )
129, 11orbi12d 690 . . . . 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 706 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  A )  ->  R  Er  X )
15 erdisj 6707 . . . . . 6  |-  ( R  Er  X  ->  ( [ x ] R  =  [ y ] R  \/  ( [ x ] R  i^i  [ y ] R )  =  (/) ) )
1614, 15syl 15 . . . . 5  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  A )  ->  ( [ x ] R  =  [ y ] R  \/  ( [ x ] R  i^i  [ y ] R )  =  (/) ) )
172, 12, 16ectocld 6726 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  C  e.  ( A /. R
) )  ->  ( [ x ] R  =  C  \/  ( [ x ] R  i^i  C )  =  (/) ) )
188, 17mpdan 649 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( [ x ] R  =  C  \/  ( [ x ] R  i^i  C )  =  (/) ) )
192, 6, 18ectocld 6726 . 2  |-  ( (
ph  /\  B  e.  ( A /. R ) )  ->  ( B  =  C  \/  ( B  i^i  C )  =  (/) ) )
201, 19mpdan 649 1  |-  ( ph  ->  ( B  =  C  \/  ( B  i^i  C )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151   (/)c0 3455    Er wer 6657   [cec 6658   /.cqs 6659
This theorem is referenced by:  qsdisj2  6737  cldsubg  17793  uninqs  25039  erprt  26741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-er 6660  df-ec 6662  df-qs 6666
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