Theorem difabs 3263

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

Step	Hyp	Ref	Expression
1		difun1 3259	. 2 |- ( A \ ( B u. B ) ) = ( ( A \ B ) \ B )
2		unidm 3143	. . 3 |- ( B u. B ) = B
3	2	difeq2i 3115	. 2 |- ( A \ ( B u. B ) ) = ( A \ B )
4	1, 3	eqtr3i 2110	1 |- ( ( A \ B ) \ B ) = ( A \ B )

Syntax hints:   = wceq 1289   \ cdif 2996   u. cun 2997

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005

This theorem is referenced by: (None)