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Theorem invdif 3715
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 3710 . 2  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  ( _V  \  ( _V  \  B ) ) )
2 ddif 3598 . . 3  |-  ( _V 
\  ( _V  \  B ) )  =  B
32difeq2i 3581 . 2  |-  ( A 
\  ( _V  \ 
( _V  \  B
) ) )  =  ( A  \  B
)
41, 3eqtri 2452 1  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1438   _Vcvv 3082    \ cdif 3434    i^i cin 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rab 2785  df-v 3084  df-dif 3440  df-in 3444
This theorem is referenced by:  indif2  3717  difundi  3726  difundir  3727  difindi  3728  difindir  3729  difdif2  3731  difun1  3734  undif1  3871  difdifdir  3884  frnsuppeq  6935  dfsup2  7962  fsets  15142
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