Theorem difabs 4269

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 4265	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = ((𝐴 ∖ 𝐵) ∖ 𝐵)
2		unidm 4123	. . 3 ⊢ (𝐵 ∪ 𝐵) = 𝐵
3	2	difeq2i 4089	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = (𝐴 ∖ 𝐵)
4	1, 3	eqtr3i 2755	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1540   ∖ cdif 3914   ∪ cun 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924

This theorem is referenced by:  axcclem  10417  lpdifsn  23037  compne  44437