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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

Proof of Theorem erdisj
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 neq0 3465 . . . 4
2 simpl 443 . . . . . . 7
3 elin 3358 . . . . . . . . . . 11
43simplbi 446 . . . . . . . . . 10
54adantl 452 . . . . . . . . 9
6 vex 2791 . . . . . . . . . 10
7 ecexr 6665 . . . . . . . . . . 11
85, 7syl 15 . . . . . . . . . 10
9 elecg 6698 . . . . . . . . . 10
106, 8, 9sylancr 644 . . . . . . . . 9
115, 10mpbid 201 . . . . . . . 8
123simprbi 450 . . . . . . . . . 10
1312adantl 452 . . . . . . . . 9
14 ecexr 6665 . . . . . . . . . . 11
1513, 14syl 15 . . . . . . . . . 10
16 elecg 6698 . . . . . . . . . 10
176, 15, 16sylancr 644 . . . . . . . . 9
1813, 17mpbid 201 . . . . . . . 8
192, 11, 18ertr4d 6679 . . . . . . 7
202, 19erthi 6706 . . . . . 6
2120ex 423 . . . . 5
2221exlimdv 1664 . . . 4
231, 22syl5bi 208 . . 3
2423orrd 367 . 2
2524orcomd 377 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wo 357   wa 358  wex 1528   wceq 1623   wcel 1684  cvv 2788   cin 3151  c0 3455   class class class wbr 4023   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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