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Theorem difsn 3755
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 3749 . . 3  |-  ( x  e.  ( B  \  { A } )  <->  ( x  e.  B  /\  x  =/=  A ) )
2 simpl 443 . . . 4  |-  ( ( x  e.  B  /\  x  =/=  A )  ->  x  e.  B )
3 eleq1 2343 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
43biimpcd 215 . . . . . . 7  |-  ( x  e.  B  ->  (
x  =  A  ->  A  e.  B )
)
54necon3bd 2483 . . . . . 6  |-  ( x  e.  B  ->  ( -.  A  e.  B  ->  x  =/=  A ) )
65com12 27 . . . . 5  |-  ( -.  A  e.  B  -> 
( x  e.  B  ->  x  =/=  A ) )
76ancld 536 . . . 4  |-  ( -.  A  e.  B  -> 
( x  e.  B  ->  ( x  e.  B  /\  x  =/=  A
) ) )
82, 7impbid2 195 . . 3  |-  ( -.  A  e.  B  -> 
( ( x  e.  B  /\  x  =/= 
A )  <->  x  e.  B ) )
91, 8syl5bb 248 . 2  |-  ( -.  A  e.  B  -> 
( x  e.  ( B  \  { A } )  <->  x  e.  B ) )
109eqrdv 2281 1  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640
This theorem is referenced by:  difsneq  3757  domdifsn  6945  domunsncan  6962  frfi  7102  infdifsn  7357  dfn2  9978  clslp  16879  xrge00  23311  nbgrassovt  28150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-sn 3646
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