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Theorem difsnb 3534
Description:  ( B  \  { A } ) equals  B if and only if  A is not a member of  B. Generalization of difsn 3529. (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 3529 . 2  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
2 neldifsnd 3526 . . . . 5  |-  ( A  e.  B  ->  -.  A  e.  ( B  \  { A } ) )
3 nelne1 2310 . . . . 5  |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  { A }
) )  ->  B  =/=  ( B  \  { A } ) )
42, 3mpdan 406 . . . 4  |-  ( A  e.  B  ->  B  =/=  ( B  \  { A } ) )
54necomd 2306 . . 3  |-  ( A  e.  B  ->  ( B  \  { A }
)  =/=  B )
65necon2bi 2275 . 2  |-  ( ( B  \  { A } )  =  B  ->  -.  A  e.  B )
71, 6impbii 121 1  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 102    = wceq 1259    e. wcel 1409    =/= wne 2220    \ cdif 2942   {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-dif 2948  df-sn 3409
This theorem is referenced by:  difsnpssim  3535
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