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Theorem difabs 3198
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 3194 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( ( A  \  B )  \  B
)
2 unidm 3083 . . 3  |-  ( B  u.  B )  =  B
32difeq2i 3056 . 2  |-  ( A 
\  ( B  u.  B ) )  =  ( A  \  B
)
41, 3eqtr3i 2062 1  |-  ( ( A  \  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1243    \ cdif 2911    u. cun 2912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rab 2312  df-v 2556  df-dif 2917  df-un 2919  df-in 2921
This theorem is referenced by: (None)
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