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Theorem difsneq 3757
 Description: equals if and only if is not a member of . Generalization of difsn 3755. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsneq

Proof of Theorem difsneq
StepHypRef Expression
1 difsn 3755 . 2
2 neldifsnd 3752 . . . . 5
3 nelne1 2535 . . . . 5
42, 3mpdan 649 . . . 4
54necomd 2529 . . 3
65necon2bi 2492 . 2
71, 6impbii 180 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 176   wceq 1623   wcel 1684   wne 2446   cdif 3149  csn 3640 This theorem is referenced by:  difsnpss  3758  incexclem  12295  mrieqv2d  13541  mreexmrid  13545  mreexexlem2d  13547  mreexexlem4d  13549  acsfiindd  14280 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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-sn 3646
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