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

Theorem invdif 3412
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 3407 . 2  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  ( _V  \  ( _V  \  B ) ) )
2 ddif 3310 . . 3  |-  ( _V 
\  ( _V  \  B ) )  =  B
32difeq2i 3293 . 2  |-  ( A 
\  ( _V  \ 
( _V  \  B
) ) )  =  ( A  \  B
)
41, 3eqtri 2305 1  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1624   _Vcvv 2790    \ cdif 3151    i^i cin 3153
This theorem is referenced by:  indif2  3414  difundi  3423  difundir  3424  difindi  3425  difindir  3426  difun1  3430  undif1  3531  difdifdir  3543  dfsup2  7191  dfsup2OLD  7192  nn0supp  10013  fsuppeq  26659
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rab 2554  df-v 2792  df-dif 3157  df-in 3161
  Copyright terms: Public domain W3C validator