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Theorem erdisj 6723
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 )  =  (/) ) )

Proof of Theorem erdisj
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 neq0 3478 . . . 4  |-  ( -.  ( [ A ] R  i^i  [ B ] R )  =  (/)  <->  E. x  x  e.  ( [ A ] R  i^i  [ B ] R ) )
2 simpl 443 . . . . . . 7  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  R  Er  X
)
3 elin 3371 . . . . . . . . . . 11  |-  ( x  e.  ( [ A ] R  i^i  [ B ] R )  <->  ( x  e.  [ A ] R  /\  x  e.  [ B ] R ) )
43simplbi 446 . . . . . . . . . 10  |-  ( x  e.  ( [ A ] R  i^i  [ B ] R )  ->  x  e.  [ A ] R
)
54adantl 452 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  x  e.  [ A ] R )
6 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
7 ecexr 6681 . . . . . . . . . . 11  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
85, 7syl 15 . . . . . . . . . 10  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A  e.  _V )
9 elecg 6714 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
106, 8, 9sylancr 644 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  ( x  e. 
[ A ] R  <->  A R x ) )
115, 10mpbid 201 . . . . . . . 8  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A R x )
123simprbi 450 . . . . . . . . . 10  |-  ( x  e.  ( [ A ] R  i^i  [ B ] R )  ->  x  e.  [ B ] R
)
1312adantl 452 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  x  e.  [ B ] R )
14 ecexr 6681 . . . . . . . . . . 11  |-  ( x  e.  [ B ] R  ->  B  e.  _V )
1513, 14syl 15 . . . . . . . . . 10  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  B  e.  _V )
16 elecg 6714 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  B  e.  _V )  ->  ( x  e.  [ B ] R  <->  B R x ) )
176, 15, 16sylancr 644 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  ( x  e. 
[ B ] R  <->  B R x ) )
1813, 17mpbid 201 . . . . . . . 8  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  B R x )
192, 11, 18ertr4d 6695 . . . . . . 7  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A R B )
202, 19erthi 6722 . . . . . 6  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  [ A ] R  =  [ B ] R )
2120ex 423 . . . . 5  |-  ( R  Er  X  ->  (
x  e.  ( [ A ] R  i^i  [ B ] R )  ->  [ A ] R  =  [ B ] R ) )
2221exlimdv 1626 . . . 4  |-  ( R  Er  X  ->  ( E. x  x  e.  ( [ A ] R  i^i  [ B ] R
)  ->  [ A ] R  =  [ B ] R ) )
231, 22syl5bi 208 . . 3  |-  ( R  Er  X  ->  ( -.  ( [ A ] R  i^i  [ B ] R )  =  (/)  ->  [ A ] R  =  [ B ] R
) )
2423orrd 367 . 2  |-  ( R  Er  X  ->  (
( [ A ] R  i^i  [ B ] R )  =  (/)  \/ 
[ A ] R  =  [ B ] R
) )
2524orcomd 377 1  |-  ( R  Er  X  ->  ( [ A ] R  =  [ B ] R  \/  ( [ A ] R  i^i  [ B ] R )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   (/)c0 3468   class class class wbr 4039    Er wer 6673   [cec 6674
This theorem is referenced by:  qsdisj  6752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-er 6676  df-ec 6678
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