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Theorem difabs 3762
Description: Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
Assertion
Ref Expression
difabs  |-  ( ( A  \  B ) 
\  B )  =  ( A  \  B
)

Proof of Theorem difabs
StepHypRef Expression
1 difun1 3758 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( ( A  \  B )  \  B
)
2 unidm 3647 . . 3  |-  ( B  u.  B )  =  B
32difeq2i 3619 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( A  \  B
)
41, 3eqtr3i 2498 1  |-  ( ( A  \  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    \ cdif 3473    u. cun 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483
This theorem is referenced by:  axcclem  8833  lpdifsn  19410  bwthOLD  19677  compne  30927
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