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Theorem difsnb 4020
Description:  ( B  \  { A } ) equals  B if and only if  A is not a member of  B. Generalization of difsn 4013. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnb  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )

Proof of Theorem difsnb
StepHypRef Expression
1 difsn 4013 . 2  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
2 neldifsnd 4008 . . . . 5  |-  ( A  e.  B  ->  -.  A  e.  ( B  \  { A } ) )
3 nelne1 2706 . . . . 5  |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  { A }
) )  ->  B  =/=  ( B  \  { A } ) )
42, 3mpdan 668 . . . 4  |-  ( A  e.  B  ->  B  =/=  ( B  \  { A } ) )
54necomd 2700 . . 3  |-  ( A  e.  B  ->  ( B  \  { A }
)  =/=  B )
65necon2bi 2662 . 2  |-  ( ( B  \  { A } )  =  B  ->  -.  A  e.  B )
71, 6impbii 188 1  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1369    e. wcel 1756    =/= wne 2611    \ cdif 3330   {csn 3882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-v 2979  df-dif 3336  df-sn 3883
This theorem is referenced by:  difsnpss  4021  incexclem  13304  mrieqv2d  14582  mreexmrid  14586  mreexexlem2d  14588  mreexexlem4d  14590  acsfiindd  15352
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