MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfin2 Unicode version

Theorem dfin2 3405
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3404. Another version is given by dfin4 3409. (Contributed by NM, 10-Jun-2004.)
Assertion
Ref Expression
dfin2  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )

Proof of Theorem dfin2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . 6  |-  x  e. 
_V
2 eldif 3162 . . . . . 6  |-  ( x  e.  ( _V  \  B )  <->  ( x  e.  _V  /\  -.  x  e.  B ) )
31, 2mpbiran 884 . . . . 5  |-  ( x  e.  ( _V  \  B )  <->  -.  x  e.  B )
43con2bii 322 . . . 4  |-  ( x  e.  B  <->  -.  x  e.  ( _V  \  B
) )
54anbi2i 675 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B ) ) )
6 eldif 3162 . . 3  |-  ( x  e.  ( A  \ 
( _V  \  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B
) ) )
75, 6bitr4i 243 . 2  |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  ( A  \ 
( _V  \  B
) ) )
87ineqri 3362 1  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    i^i cin 3151
This theorem is referenced by:  dfun3  3407  dfin3  3408  invdif  3410  difundi  3421  difindi  3423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159
  Copyright terms: Public domain W3C validator