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Theorem erdisj 6943
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 3630 . . . 4  |-  ( -.  ( [ A ] R  i^i  [ B ] R )  =  (/)  <->  E. x  x  e.  ( [ A ] R  i^i  [ B ] R ) )
2 simpl 444 . . . . . . 7  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  R  Er  X
)
3 elin 3522 . . . . . . . . . . 11  |-  ( x  e.  ( [ A ] R  i^i  [ B ] R )  <->  ( x  e.  [ A ] R  /\  x  e.  [ B ] R ) )
43simplbi 447 . . . . . . . . . 10  |-  ( x  e.  ( [ A ] R  i^i  [ B ] R )  ->  x  e.  [ A ] R
)
54adantl 453 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  x  e.  [ A ] R )
6 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
7 ecexr 6901 . . . . . . . . . . 11  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
85, 7syl 16 . . . . . . . . . 10  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A  e.  _V )
9 elecg 6934 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
106, 8, 9sylancr 645 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  ( x  e. 
[ A ] R  <->  A R x ) )
115, 10mpbid 202 . . . . . . . 8  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A R x )
123simprbi 451 . . . . . . . . . 10  |-  ( x  e.  ( [ A ] R  i^i  [ B ] R )  ->  x  e.  [ B ] R
)
1312adantl 453 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  x  e.  [ B ] R )
14 ecexr 6901 . . . . . . . . . . 11  |-  ( x  e.  [ B ] R  ->  B  e.  _V )
1513, 14syl 16 . . . . . . . . . 10  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  B  e.  _V )
16 elecg 6934 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  B  e.  _V )  ->  ( x  e.  [ B ] R  <->  B R x ) )
176, 15, 16sylancr 645 . . . . . . . . 9  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  ( x  e. 
[ B ] R  <->  B R x ) )
1813, 17mpbid 202 . . . . . . . 8  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  B R x )
192, 11, 18ertr4d 6915 . . . . . . 7  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  A R B )
202, 19erthi 6942 . . . . . 6  |-  ( ( R  Er  X  /\  x  e.  ( [ A ] R  i^i  [ B ] R ) )  ->  [ A ] R  =  [ B ] R )
2120ex 424 . . . . 5  |-  ( R  Er  X  ->  (
x  e.  ( [ A ] R  i^i  [ B ] R )  ->  [ A ] R  =  [ B ] R ) )
2221exlimdv 1646 . . . 4  |-  ( R  Er  X  ->  ( E. x  x  e.  ( [ A ] R  i^i  [ B ] R
)  ->  [ A ] R  =  [ B ] R ) )
231, 22syl5bi 209 . . 3  |-  ( R  Er  X  ->  ( -.  ( [ A ] R  i^i  [ B ] R )  =  (/)  ->  [ A ] R  =  [ B ] R
) )
2423orrd 368 . 2  |-  ( R  Er  X  ->  (
( [ A ] R  i^i  [ B ] R )  =  (/)  \/ 
[ A ] R  =  [ B ] R
) )
2524orcomd 378 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 177    \/ wo 358    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311   (/)c0 3620   class class class wbr 4204    Er wer 6893   [cec 6894
This theorem is referenced by:  qsdisj  6972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-er 6896  df-ec 6898
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