Theorem difabs 3401

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 3397	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = ((𝐴 ∖ 𝐵) ∖ 𝐵)
2		unidm 3280	. . 3 ⊢ (𝐵 ∪ 𝐵) = 𝐵
3	2	difeq2i 3252	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = (𝐴 ∖ 𝐵)
4	1, 3	eqtr3i 2200	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1353   ∖ cdif 3128   ∪ cun 3129

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137

This theorem is referenced by: (None)