Theorem difabs 3519

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 3515	. 2 ⊢ (A ∖ (B ∪ B)) = ((A ∖ B) ∖ B)
2		unidm 3408	. . 3 ⊢ (B ∪ B) = B
3	2	difeq2i 3383	. 2 ⊢ (A ∖ (B ∪ B)) = (A ∖ B)
4	1, 3	eqtr3i 2375	1 ⊢ ((A ∖ B) ∖ B) = (A ∖ B)

Syntax hints:   = wceq 1642   ∖ cdif 3207   ∪ cun 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216

This theorem is referenced by: (None)