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Theorem invdif 3589
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 3584 . 2  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  ( _V  \  ( _V  \  B ) ) )
2 ddif 3486 . . 3  |-  ( _V 
\  ( _V  \  B ) )  =  B
32difeq2i 3469 . 2  |-  ( A 
\  ( _V  \ 
( _V  \  B
) ) )  =  ( A  \  B
)
41, 3eqtri 2461 1  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   _Vcvv 2970    \ cdif 3323    i^i cin 3325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rab 2722  df-v 2972  df-dif 3329  df-in 3333
This theorem is referenced by:  indif2  3591  difundi  3600  difundir  3601  difindi  3602  difindir  3603  difdif2  3605  difun1  3608  undif1  3752  difdifdir  3764  frnsuppeq  6700  dfsup2  7690  dfsup2OLD  7691  nn0suppOLD  10632  fsets  14198  fsuppeq  29447
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