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Theorem invdif 3224
Description: Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
invdif  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)

Proof of Theorem invdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2615 . . . . 5  |-  x  e. 
_V
2 eldif 2993 . . . . 5  |-  ( x  e.  ( _V  \  B )  <->  ( x  e.  _V  /\  -.  x  e.  B ) )
31, 2mpbiran 882 . . . 4  |-  ( x  e.  ( _V  \  B )  <->  -.  x  e.  B )
43anbi2i 445 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( _V  \  B ) )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 elin 3167 . . 3  |-  ( x  e.  ( A  i^i  ( _V  \  B ) )  <->  ( x  e.  A  /\  x  e.  ( _V  \  B
) ) )
6 eldif 2993 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
74, 5, 63bitr4i 210 . 2  |-  ( x  e.  ( A  i^i  ( _V  \  B ) )  <->  x  e.  ( A  \  B ) )
87eqriv 2080 1  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2612    \ cdif 2981    i^i cin 2983
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-in 2990
This theorem is referenced by:  indif2  3226  difundir  3235  difindir  3237  difdif2ss  3239  difun1  3242  difdifdirss  3348  nn0supp  8615
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