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Theorem difsnb 3580
Description:  ( B  \  { A } ) equals  B if and only if  A is not a member of  B. Generalization of difsn 3574. (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 3574 . 2  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
2 neldifsnd 3571 . . . . 5  |-  ( A  e.  B  ->  -.  A  e.  ( B  \  { A } ) )
3 nelne1 2345 . . . . 5  |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  { A }
) )  ->  B  =/=  ( B  \  { A } ) )
42, 3mpdan 412 . . . 4  |-  ( A  e.  B  ->  B  =/=  ( B  \  { A } ) )
54necomd 2341 . . 3  |-  ( A  e.  B  ->  ( B  \  { A }
)  =/=  B )
65necon2bi 2310 . 2  |-  ( ( B  \  { A } )  =  B  ->  -.  A  e.  B )
71, 6impbii 124 1  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103    = wceq 1289    e. wcel 1438    =/= wne 2255    \ cdif 2996   {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 3001  df-sn 3452
This theorem is referenced by: (None)
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