Theorem difabs 3851

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 3846	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = ((𝐴 ∖ 𝐵) ∖ 𝐵)
2		unidm 3718	. . 3 ⊢ (𝐵 ∪ 𝐵) = 𝐵
3	2	difeq2i 3687	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = (𝐴 ∖ 𝐵)
4	1, 3	eqtr3i 2634	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1475   ∖ cdif 3537   ∪ cun 3538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547

This theorem is referenced by:  axcclem  9162  lpdifsn  20757  compne  37665