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Theorem qsdisj 7406
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 2457 . . 3  |-  ( A /. R )  =  ( A /. R
)
3 eqeq1 2461 . . . 4  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =  C  <->  B  =  C
) )
4 ineq1 3689 . . . . 5  |-  ( [ x ] R  =  B  ->  ( [
x ] R  i^i  C )  =  ( B  i^i  C ) )
54eqeq1d 2459 . . . 4  |-  ( [ x ] R  =  B  ->  ( ( [ x ] R  i^i  C )  =  (/)  <->  ( B  i^i  C )  =  (/) ) )
63, 5orbi12d 709 . . 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 465 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  ( A /. R
) )
9 eqeq2 2472 . . . . . 6  |-  ( [ y ] R  =  C  ->  ( [
x ] R  =  [ y ] R  <->  [ x ] R  =  C ) )
10 ineq2 3690 . . . . . . 7  |-  ( [ y ] R  =  C  ->  ( [
x ] R  i^i  [ y ] R )  =  ( [ x ] R  i^i  C ) )
1110eqeq1d 2459 . . . . . 6  |-  ( [ y ] R  =  C  ->  ( ( [ x ] R  i^i  [ y ] R
)  =  (/)  <->  ( [
x ] R  i^i  C )  =  (/) ) )
129, 11orbi12d 709 . . . . 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 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  A )  ->  R  Er  X )
15 erdisj 7377 . . . . . 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 7396 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  C  e.  ( A /. R
) )  ->  ( [ x ] R  =  C  \/  ( [ x ] R  i^i  C )  =  (/) ) )
188, 17mpdan 668 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( [ x ] R  =  C  \/  ( [ x ] R  i^i  C )  =  (/) ) )
192, 6, 18ectocld 7396 . 2  |-  ( (
ph  /\  B  e.  ( A /. R ) )  ->  ( B  =  C  \/  ( B  i^i  C )  =  (/) ) )
201, 19mpdan 668 1  |-  ( ph  ->  ( B  =  C  \/  ( B  i^i  C )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    i^i cin 3470   (/)c0 3793    Er wer 7326   [cec 7327   /.cqs 7328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-er 7329  df-ec 7331  df-qs 7335
This theorem is referenced by:  qsdisj2  7407  uniinqs  7409  cldsubg  20734  erprt  30776
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