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Theorem erdisj 6707
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 3465 . . . 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 3358 . . . . . . . . . . 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 2791 . . . . . . . . . 10  |-  x  e. 
_V
7 ecexr 6665 . . . . . . . . . . 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 6698 . . . . . . . . . 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 6665 . . . . . . . . . . 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 6698 . . . . . . . . . 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 6679 . . . . . . 7  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A R B )
202, 19erthi 6706 . . . . . 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 1664 . . . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   (/)c0 3455   class class class wbr 4023    Er wer 6657   [cec 6658
This theorem is referenced by:  qsdisj  6736
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
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