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Theorem invdif 3369
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 2733 . . . . 5  |-  x  e. 
_V
2 eldif 3130 . . . . 5  |-  ( x  e.  ( _V  \  B )  <->  ( x  e.  _V  /\  -.  x  e.  B ) )
31, 2mpbiran 935 . . . 4  |-  ( x  e.  ( _V  \  B )  <->  -.  x  e.  B )
43anbi2i 454 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( _V  \  B ) )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 elin 3310 . . 3  |-  ( x  e.  ( A  i^i  ( _V  \  B ) )  <->  ( x  e.  A  /\  x  e.  ( _V  \  B
) ) )
6 eldif 3130 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
74, 5, 63bitr4i 211 . 2  |-  ( x  e.  ( A  i^i  ( _V  \  B ) )  <->  x  e.  ( A  \  B ) )
87eqriv 2167 1  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1348    e. wcel 2141   _Vcvv 2730    \ cdif 3118    i^i cin 3120
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127
This theorem is referenced by:  indif2  3371  difundir  3380  difindir  3382  difdif2ss  3384  difun1  3387  difdifdirss  3499  nn0supp  9187
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