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Theorem difsnb 3497
Description:  \  { } equals if and only if is not a member of . Generalization of difsn 3492. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnb 
\  { }

Proof of Theorem difsnb
StepHypRef Expression
1 difsn 3492 . 2  \  { }
2 neldifsnd 3489 . . . . 5  \  { }
3 nelne1 2289 . . . . 5 
\  { }  =/=  \  { }
42, 3mpdan 398 . . . 4  =/=  \  { }
54necomd 2285 . . 3  \  { }  =/=
65necon2bi 2254 . 2  \  { }
71, 6impbii 117 1 
\  { }
Colors of variables: wff set class
Syntax hints:   wn 3   wb 98   wceq 1242   wcel 1390    =/= wne 2201    \ cdif 2908   {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-sn 3373
This theorem is referenced by:  difsnpssim  3498
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