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Theorem difdif 3098
Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif  |-  ( A 
\  ( B  \  A ) )  =  A

Proof of Theorem difdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl 107 . . 3  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  \  A ) )  ->  x  e.  A
)
2 pm4.45im 327 . . . 4  |-  ( x  e.  A  <->  ( x  e.  A  /\  (
x  e.  B  ->  x  e.  A )
) )
3 imanim 819 . . . . . 6  |-  ( ( x  e.  B  ->  x  e.  A )  ->  -.  ( x  e.  B  /\  -.  x  e.  A ) )
4 eldif 2983 . . . . . 6  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
53, 4sylnibr 635 . . . . 5  |-  ( ( x  e.  B  ->  x  e.  A )  ->  -.  x  e.  ( B  \  A ) )
65anim2i 334 . . . 4  |-  ( ( x  e.  A  /\  ( x  e.  B  ->  x  e.  A ) )  ->  ( x  e.  A  /\  -.  x  e.  ( B  \  A
) ) )
72, 6sylbi 119 . . 3  |-  ( x  e.  A  ->  (
x  e.  A  /\  -.  x  e.  ( B  \  A ) ) )
81, 7impbii 124 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  \  A ) )  <-> 
x  e.  A )
98difeqri 3093 1  |-  ( A 
\  ( B  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    \ cdif 2971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976
This theorem is referenced by:  dif0  3321
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