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Theorem indif 3320
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 563 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 elin 3260 . . . 4  |-  ( x  e.  ( A  i^i  ( A  \  B ) )  <->  ( x  e.  A  /\  x  e.  ( A  \  B
) ) )
3 eldif 3081 . . . . 5  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
43anbi2i 453 . . . 4  |-  ( ( x  e.  A  /\  x  e.  ( A  \  B ) )  <->  ( x  e.  A  /\  (
x  e.  A  /\  -.  x  e.  B
) ) )
52, 4bitri 183 . . 3  |-  ( x  e.  ( A  i^i  ( A  \  B ) )  <->  ( x  e.  A  /\  ( x  e.  A  /\  -.  x  e.  B )
) )
61, 5, 33bitr4i 211 . 2  |-  ( x  e.  ( A  i^i  ( A  \  B ) )  <->  x  e.  ( A  \  B ) )
76eqriv 2137 1  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1332    e. wcel 1481    \ cdif 3069    i^i cin 3071
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-dif 3074  df-in 3078
This theorem is referenced by:  resdif  5393
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