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Theorem difsnb 4169
Description:  ( B  \  { A } ) equals  B if and only if  A is not a member of  B. Generalization of difsn 4161. (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 4161 . 2  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
2 neldifsnd 4155 . . . . 5  |-  ( A  e.  B  ->  -.  A  e.  ( B  \  { A } ) )
3 nelne1 2796 . . . . 5  |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  { A }
) )  ->  B  =/=  ( B  \  { A } ) )
42, 3mpdan 668 . . . 4  |-  ( A  e.  B  ->  B  =/=  ( B  \  { A } ) )
54necomd 2738 . . 3  |-  ( A  e.  B  ->  ( B  \  { A }
)  =/=  B )
65necon2bi 2704 . 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 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   {csn 4027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-sn 4028
This theorem is referenced by:  difsnpss  4170  incexclem  13604  mrieqv2d  14887  mreexmrid  14891  mreexexlem2d  14893  mreexexlem4d  14895  acsfiindd  15657
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