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Theorem difsn 4111
Description: An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
difsn  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )

Proof of Theorem difsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldifsn 4103 . . 3  |-  ( x  e.  ( B  \  { A } )  <->  ( x  e.  B  /\  x  =/=  A ) )
2 simpl 457 . . . 4  |-  ( ( x  e.  B  /\  x  =/=  A )  ->  x  e.  B )
3 eleq1 2524 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
43biimpcd 224 . . . . . . 7  |-  ( x  e.  B  ->  (
x  =  A  ->  A  e.  B )
)
54necon3bd 2661 . . . . . 6  |-  ( x  e.  B  ->  ( -.  A  e.  B  ->  x  =/=  A ) )
65com12 31 . . . . 5  |-  ( -.  A  e.  B  -> 
( x  e.  B  ->  x  =/=  A ) )
76ancld 553 . . . 4  |-  ( -.  A  e.  B  -> 
( x  e.  B  ->  ( x  e.  B  /\  x  =/=  A
) ) )
82, 7impbid2 204 . . 3  |-  ( -.  A  e.  B  -> 
( ( x  e.  B  /\  x  =/= 
A )  <->  x  e.  B ) )
91, 8syl5bb 257 . 2  |-  ( -.  A  e.  B  -> 
( x  e.  ( B  \  { A } )  <->  x  e.  B ) )
109eqrdv 2449 1  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645    \ cdif 3428   {csn 3980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-v 3074  df-dif 3434  df-sn 3981
This theorem is referenced by:  difsnb  4118  difsnexi  6489  domdifsn  7499  domunsncan  7516  frfi  7663  infdifsn  7968  dfn2  10698  clslp  18879  nbgrassovt  23493  xrge00  26287
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