Theorem difdifdirss 3581

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 3472	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
2		invdif 3451	. . . . 5 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
3	1, 2	eqtr4i 2255	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
4		un0 3530	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
5	3, 4	eqtr4i 2255	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6		indi 3456	. . . 4 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
7		disjdif 3569	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅
8		incom 3401	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐶)
9	7, 8	eqtr3i 2254	. . . . 5 ⊢ ∅ = ((𝐴 ∖ 𝐶) ∩ 𝐶)
10	9	uneq2i 3360	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
11	6, 10	eqtr4i 2255	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
12	5, 11	eqtr4i 2255	. 2 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13		ddifss 3447	. . . . . 6 ⊢ 𝐶 ⊆ (V ∖ (V ∖ 𝐶))
14		unss2 3380	. . . . . 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 3468	. . . . . 6 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵 ∩ (V ∖ 𝐶)))
17		invdif 3451	. . . . . . 7 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
18	17	difeq2i 3324	. . . . . 6 ⊢ (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵 ∖ 𝐶))
19	16, 18	sseqtri 3262	. . . . 5 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) ⊆ (V ∖ (𝐵 ∖ 𝐶))
20	15, 19	sstri 3237	. . . 4 ⊢ ((V ∖ 𝐵) ∪ 𝐶) ⊆ (V ∖ (𝐵 ∖ 𝐶))
21		sslin 3435	. . . 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 3451	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
24	22, 23	sseqtri 3262	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) ⊆ ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
25	12, 24	eqsstri 3260	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) ⊆ ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))

Syntax hints:  Vcvv 2803   ∖ cdif 3198   ∪ cun 3199   ∩ cin 3200   ⊆ wss 3201  ∅c0 3496

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497

This theorem is referenced by: (None)