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Theorem difdif 3260
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 109 . . 3  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  \  A ) )  ->  x  e.  A
)
2 pm4.45im 334 . . . 4  |-  ( x  e.  A  <->  ( x  e.  A  /\  (
x  e.  B  ->  x  e.  A )
) )
3 imanim 688 . . . . . 6  |-  ( ( x  e.  B  ->  x  e.  A )  ->  -.  ( x  e.  B  /\  -.  x  e.  A ) )
4 eldif 3138 . . . . . 6  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
53, 4sylnibr 677 . . . . 5  |-  ( ( x  e.  B  ->  x  e.  A )  ->  -.  x  e.  ( B  \  A ) )
65anim2i 342 . . . 4  |-  ( ( x  e.  A  /\  ( x  e.  B  ->  x  e.  A ) )  ->  ( x  e.  A  /\  -.  x  e.  ( B  \  A
) ) )
72, 6sylbi 121 . . 3  |-  ( x  e.  A  ->  (
x  e.  A  /\  -.  x  e.  ( B  \  A ) ) )
81, 7impbii 126 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  \  A ) )  <-> 
x  e.  A )
98difeqri 3255 1  |-  ( A 
\  ( B  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    \ cdif 3126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131
This theorem is referenced by:  dif0  3493
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