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Theorem difsn 3652
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 3645 . . 3  |-  ( x  e.  ( B  \  { A } )  <->  ( x  e.  B  /\  x  =/=  A ) )
2 simpl 108 . . . 4  |-  ( ( x  e.  B  /\  x  =/=  A )  ->  x  e.  B )
3 eleq1 2200 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
43biimpcd 158 . . . . . . 7  |-  ( x  e.  B  ->  (
x  =  A  ->  A  e.  B )
)
54necon3bd 2349 . . . . . 6  |-  ( x  e.  B  ->  ( -.  A  e.  B  ->  x  =/=  A ) )
65com12 30 . . . . 5  |-  ( -.  A  e.  B  -> 
( x  e.  B  ->  x  =/=  A ) )
76ancld 323 . . . 4  |-  ( -.  A  e.  B  -> 
( x  e.  B  ->  ( x  e.  B  /\  x  =/=  A
) ) )
82, 7impbid2 142 . . 3  |-  ( -.  A  e.  B  -> 
( ( x  e.  B  /\  x  =/= 
A )  <->  x  e.  B ) )
91, 8syl5bb 191 . 2  |-  ( -.  A  e.  B  -> 
( x  e.  ( B  \  { A } )  <->  x  e.  B ) )
109eqrdv 2135 1  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    =/= wne 2306    \ cdif 3063   {csn 3522
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-sn 3528
This theorem is referenced by:  difsnb  3658  fisseneq  6813  dfn2  8983
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