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

Theorem invdif 3410
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
StepHypRef Expression
1 dfin2 3405 . 2  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  ( _V  \  ( _V  \  B ) ) )
2 ddif 3308 . . 3  |-  ( _V 
\  ( _V  \  B ) )  =  B
32difeq2i 3291 . 2  |-  ( A 
\  ( _V  \ 
( _V  \  B
) ) )  =  ( A  \  B
)
41, 3eqtri 2303 1  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    \ cdif 3149    i^i cin 3151
This theorem is referenced by:  indif2  3412  difundi  3421  difundir  3422  difindi  3423  difindir  3424  difun1  3428  undif1  3529  difdifdir  3541  dfsup2  7195  dfsup2OLD  7196  nn0supp  10017  fsuppeq  27259
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-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-in 3159
  Copyright terms: Public domain W3C validator