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Theorem erdisj 7378
 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

Proof of Theorem erdisj
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 neq0 3795 . . . 4
2 simpl 457 . . . . . . 7
3 elin 3686 . . . . . . . . . . 11
43simplbi 460 . . . . . . . . . 10
54adantl 466 . . . . . . . . 9
6 vex 3112 . . . . . . . . . 10
7 ecexr 7335 . . . . . . . . . . 11
85, 7syl 16 . . . . . . . . . 10
9 elecg 7369 . . . . . . . . . 10
106, 8, 9sylancr 663 . . . . . . . . 9
115, 10mpbid 210 . . . . . . . 8
123simprbi 464 . . . . . . . . . 10
1312adantl 466 . . . . . . . . 9
14 ecexr 7335 . . . . . . . . . . 11
1513, 14syl 16 . . . . . . . . . 10
16 elecg 7369 . . . . . . . . . 10
176, 15, 16sylancr 663 . . . . . . . . 9
1813, 17mpbid 210 . . . . . . . 8
192, 11, 18ertr4d 7349 . . . . . . 7
202, 19erthi 7377 . . . . . 6
2120ex 434 . . . . 5
2221exlimdv 1724 . . . 4
231, 22syl5bi 217 . . 3
2423orrd 378 . 2
2524orcomd 388 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  i^icin 3474   c0 3784   class class class wbr 4452  Erwer 7327  [cec 7328 This theorem is referenced by:  qsdisj  7407 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-er 7330  df-ec 7332
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