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Theorem difsnb 3633
Description:  ( B  \  { A } ) equals  B if and only if  A is not a member of  B. Generalization of difsn 3627. (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 3627 . 2  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
2 neldifsnd 3624 . . . . 5  |-  ( A  e.  B  ->  -.  A  e.  ( B  \  { A } ) )
3 nelne1 2375 . . . . 5  |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  { A }
) )  ->  B  =/=  ( B  \  { A } ) )
42, 3mpdan 417 . . . 4  |-  ( A  e.  B  ->  B  =/=  ( B  \  { A } ) )
54necomd 2371 . . 3  |-  ( A  e.  B  ->  ( B  \  { A }
)  =/=  B )
65necon2bi 2340 . 2  |-  ( ( B  \  { A } )  =  B  ->  -.  A  e.  B )
71, 6impbii 125 1  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104    = wceq 1316    e. wcel 1465    =/= wne 2285    \ cdif 3038   {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-v 2662  df-dif 3043  df-sn 3503
This theorem is referenced by: (None)
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