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Theorem erdisj 6702
Description: Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
erdisj  |-  ( R  Er  X  ->  ( [ A ] R  =  [ B ] R  \/  ( [ A ] R  i^i  [ B ] R )  =  (/) ) )
Dummy variable  x is distinct from all other variables.

Proof of Theorem erdisj
StepHypRef Expression
1 neq0 3466 . . . 4  |-  ( -.  ( [ A ] R  i^i  [ B ] R )  =  (/)  <->  E. x  x  e.  ( [ A ] R  i^i  [ B ] R ) )
2 simpl 445 . . . . . . 7  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  R  Er  X
)
3 elin 3359 . . . . . . . . . . 11  |-  ( x  e.  ( [ A ] R  i^i  [ B ] R )  <->  ( x  e.  [ A ] R  /\  x  e.  [ B ] R ) )
43simplbi 448 . . . . . . . . . 10  |-  ( x  e.  ( [ A ] R  i^i  [ B ] R )  ->  x  e.  [ A ] R
)
54adantl 454 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  x  e.  [ A ] R )
6 vex 2792 . . . . . . . . . 10  |-  x  e. 
_V
7 ecexr 6660 . . . . . . . . . . 11  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
85, 7syl 17 . . . . . . . . . 10  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A  e.  _V )
9 elecg 6693 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
106, 8, 9sylancr 646 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  ( x  e. 
[ A ] R  <->  A R x ) )
115, 10mpbid 203 . . . . . . . 8  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A R x )
123simprbi 452 . . . . . . . . . 10  |-  ( x  e.  ( [ A ] R  i^i  [ B ] R )  ->  x  e.  [ B ] R
)
1312adantl 454 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  x  e.  [ B ] R )
14 ecexr 6660 . . . . . . . . . . 11  |-  ( x  e.  [ B ] R  ->  B  e.  _V )
1513, 14syl 17 . . . . . . . . . 10  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  B  e.  _V )
16 elecg 6693 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  B  e.  _V )  ->  ( x  e.  [ B ] R  <->  B R x ) )
176, 15, 16sylancr 646 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  ( x  e. 
[ B ] R  <->  B R x ) )
1813, 17mpbid 203 . . . . . . . 8  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  B R x )
192, 11, 18ertr4d 6674 . . . . . . 7  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A R B )
202, 19erthi 6701 . . . . . 6  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  [ A ] R  =  [ B ] R )
2120ex 425 . . . . 5  |-  ( R  Er  X  ->  (
x  e.  ( [ A ] R  i^i  [ B ] R )  ->  [ A ] R  =  [ B ] R ) )
2221exlimdv 1665 . . . 4  |-  ( R  Er  X  ->  ( E. x  x  e.  ( [ A ] R  i^i  [ B ] R
)  ->  [ A ] R  =  [ B ] R ) )
231, 22syl5bi 210 . . 3  |-  ( R  Er  X  ->  ( -.  ( [ A ] R  i^i  [ B ] R )  =  (/)  ->  [ A ] R  =  [ B ] R
) )
2423orrd 369 . 2  |-  ( R  Er  X  ->  (
( [ A ] R  i^i  [ B ] R )  =  (/)  \/ 
[ A ] R  =  [ B ] R
) )
2524orcomd 379 1  |-  ( R  Er  X  ->  ( [ A ] R  =  [ B ] R  \/  ( [ A ] R  i^i  [ B ] R )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   _Vcvv 2789    i^i cin 3152   (/)c0 3456   class class class wbr 4024    Er wer 6652   [cec 6653
This theorem is referenced by:  qsdisj  6731
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-er 6655  df-ec 6657
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