Theorem difabs 3441

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 3437	. 2 |- ( A \ ( B u. B ) ) = ( ( A \ B ) \ B )
2		unidm 3320	. . 3 |- ( B u. B ) = B
3	2	difeq2i 3292	. 2 |- ( A \ ( B u. B ) ) = ( A \ B )
4	1, 3	eqtr3i 2229	1 |- ( ( A \ B ) \ B ) = ( A \ B )

Syntax hints:   = wceq 1373   \ cdif 3167   u. cun 3168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176

This theorem is referenced by: (None)