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Theorem qsdisj 7176
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 2442 . . 3  |-  ( A /. R )  =  ( A /. R
)
3 eqeq1 2448 . . . 4  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =  C  <->  B  =  C
) )
4 ineq1 3544 . . . . 5  |-  ( [ x ] R  =  B  ->  ( [
x ] R  i^i  C )  =  ( B  i^i  C ) )
54eqeq1d 2450 . . . 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 2451 . . . . . 6  |-  ( [ y ] R  =  C  ->  ( [
x ] R  =  [ y ] R  <->  [ x ] R  =  C ) )
10 ineq2 3545 . . . . . . 7  |-  ( [ y ] R  =  C  ->  ( [
x ] R  i^i  [ y ] R )  =  ( [ x ] R  i^i  C ) )
1110eqeq1d 2450 . . . . . 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 7147 . . . . . 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 7166 . . . 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 7166 . 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 1369    e. wcel 1756    i^i cin 3326   (/)c0 3636    Er wer 7097   [cec 7098   /.cqs 7099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-er 7100  df-ec 7102  df-qs 7106
This theorem is referenced by:  qsdisj2  7177  uniinqs  7179  cldsubg  19680  erprt  29016
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