Theorem difabs 3471

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 3467	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = ((𝐴 ∖ 𝐵) ∖ 𝐵)
2		unidm 3350	. . 3 ⊢ (𝐵 ∪ 𝐵) = 𝐵
3	2	difeq2i 3322	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = (𝐴 ∖ 𝐵)
4	1, 3	eqtr3i 2254	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1397   ∖ cdif 3197   ∪ cun 3198

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206

This theorem is referenced by: (None)