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Theorem indif 3378
Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
indif  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)

Proof of Theorem indif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 anabs5 573 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 elin 3318 . . . 4  |-  ( x  e.  ( A  i^i  ( A  \  B ) )  <->  ( x  e.  A  /\  x  e.  ( A  \  B
) ) )
3 eldif 3138 . . . . 5  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
43anbi2i 457 . . . 4  |-  ( ( x  e.  A  /\  x  e.  ( A  \  B ) )  <->  ( x  e.  A  /\  (
x  e.  A  /\  -.  x  e.  B
) ) )
52, 4bitri 184 . . 3  |-  ( x  e.  ( A  i^i  ( A  \  B ) )  <->  ( x  e.  A  /\  ( x  e.  A  /\  -.  x  e.  B )
) )
61, 5, 33bitr4i 212 . 2  |-  ( x  e.  ( A  i^i  ( A  \  B ) )  <->  x  e.  ( A  \  B ) )
76eqriv 2174 1  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    = wceq 1353    e. wcel 2148    \ cdif 3126    i^i cin 3128
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  df-in 3135
This theorem is referenced by:  resdif  5478
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