ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  invdif Unicode version

Theorem invdif 3392
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 2755 . . . . 5  |-  x  e. 
_V
2 eldif 3153 . . . . 5  |-  ( x  e.  ( _V  \  B )  <->  ( x  e.  _V  /\  -.  x  e.  B ) )
31, 2mpbiran 942 . . . 4  |-  ( x  e.  ( _V  \  B )  <->  -.  x  e.  B )
43anbi2i 457 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( _V  \  B ) )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 elin 3333 . . 3  |-  ( x  e.  ( A  i^i  ( _V  \  B ) )  <->  ( x  e.  A  /\  x  e.  ( _V  \  B
) ) )
6 eldif 3153 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
74, 5, 63bitr4i 212 . 2  |-  ( x  e.  ( A  i^i  ( _V  \  B ) )  <->  x  e.  ( A  \  B ) )
87eqriv 2186 1  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752    \ cdif 3141    i^i cin 3143
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150
This theorem is referenced by:  indif2  3394  difundir  3403  difindir  3405  difdif2ss  3407  difun1  3410  difdifdirss  3522  nn0supp  9257
  Copyright terms: Public domain W3C validator