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Theorem difabs 3310
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 3306 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( ( A  \  B )  \  B
)
2 unidm 3189 . . 3  |-  ( B  u.  B )  =  B
32difeq2i 3161 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( A  \  B
)
41, 3eqtr3i 2140 1  |-  ( ( A  \  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1316    \ cdif 3038    u. cun 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047
This theorem is referenced by: (None)
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