Theorem difdifdirss 3576

Description: Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
difdifdirss	⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) ⊆ ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))

Proof of Theorem difdifdirss

Step	Hyp	Ref	Expression
1		dif32 3467	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
2		invdif 3446	. . . . 5 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
3	1, 2	eqtr4i 2253	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
4		un0 3525	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
5	3, 4	eqtr4i 2253	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6		indi 3451	. . . 4 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
7		disjdif 3564	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅
8		incom 3396	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐶)
9	7, 8	eqtr3i 2252	. . . . 5 ⊢ ∅ = ((𝐴 ∖ 𝐶) ∩ 𝐶)
10	9	uneq2i 3355	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
11	6, 10	eqtr4i 2253	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
12	5, 11	eqtr4i 2253	. 2 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13		ddifss 3442	. . . . . 6 ⊢ 𝐶 ⊆ (V ∖ (V ∖ 𝐶))
14		unss2 3375	. . . . . 6 ⊢ (𝐶 ⊆ (V ∖ (V ∖ 𝐶)) → ((V ∖ 𝐵) ∪ 𝐶) ⊆ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))))
15	13, 14	ax-mp 5	. . . . 5 ⊢ ((V ∖ 𝐵) ∪ 𝐶) ⊆ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶)))
16		indmss 3463	. . . . . 6 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵 ∩ (V ∖ 𝐶)))
17		invdif 3446	. . . . . . 7 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
18	17	difeq2i 3319	. . . . . 6 ⊢ (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵 ∖ 𝐶))
19	16, 18	sseqtri 3258	. . . . 5 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵 ∖ 𝐶))
20	15, 19	sstri 3233	. . . 4 ⊢ ((V ∖ 𝐵) ∪ 𝐶) ⊆ (V ∖ (𝐵 ∖ 𝐶))
21		sslin 3430	. . . 4 ⊢ (((V ∖ 𝐵) ∪ 𝐶) ⊆ (V ∖ (𝐵 ∖ 𝐶)) → ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶))))
22	20, 21	ax-mp 5	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶)))
23		invdif 3446	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
24	22, 23	sseqtri 3258	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
25	12, 24	eqsstri 3256	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) ⊆ ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))

Syntax hints:  Vcvv 2799   ∖ cdif 3194   ∪ cun 3195   ∩ cin 3196   ⊆ wss 3197  ∅c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492

This theorem is referenced by: (None)