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Theorem difsn 4005
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 3997 . . 3  |-  ( x  e.  ( B  \  { A } )  <->  ( x  e.  B  /\  x  =/=  A ) )
2 simpl 454 . . . 4  |-  ( ( x  e.  B  /\  x  =/=  A )  ->  x  e.  B )
3 eleq1 2501 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
43biimpcd 224 . . . . . . 7  |-  ( x  e.  B  ->  (
x  =  A  ->  A  e.  B )
)
54necon3bd 2643 . . . . . 6  |-  ( x  e.  B  ->  ( -.  A  e.  B  ->  x  =/=  A ) )
65com12 31 . . . . 5  |-  ( -.  A  e.  B  -> 
( x  e.  B  ->  x  =/=  A ) )
76ancld 550 . . . 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 2439 1  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322   {csn 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-v 2972  df-dif 3328  df-sn 3875
This theorem is referenced by:  difsnb  4012  difsnexi  6383  domdifsn  7390  domunsncan  7407  frfi  7553  infdifsn  7858  dfn2  10588  clslp  18711  nbgrassovt  23281  xrge00  26080
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