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Theorem difabs 3597
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 3593 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( ( A  \  B )  \  B
)
2 unidm 3482 . . 3  |-  ( B  u.  B )  =  B
32difeq2i 3454 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( A  \  B
)
41, 3eqtr3i 2457 1  |-  ( ( A  \  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    \ cdif 3309    u. cun 3310
This theorem is referenced by:  axcclem  8329  lpdifsn  17199  bwth  17465  compne  27610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319
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