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Theorem difabs 3541
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 3537 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( ( A  \  B )  \  B
)
2 unidm 3426 . . 3  |-  ( B  u.  B )  =  B
32difeq2i 3398 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( A  \  B
)
41, 3eqtr3i 2402 1  |-  ( ( A  \  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    \ cdif 3253    u. cun 3254
This theorem is referenced by:  axcclem  8263  lpdifsn  17123  compne  27304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263
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