Theorem difdifdir 3638

Description: Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)

Assertion

Ref	Expression
difdifdir	⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∖ (B ∖ C))

Proof of Theorem difdifdir

Step	Hyp	Ref	Expression
1		dif32 3518	. . . . 5 ⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∖ B)
2		invdif 3497	. . . . 5 ⊢ ((A ∖ C) ∩ (V ∖ B)) = ((A ∖ C) ∖ B)
3	1, 2	eqtr4i 2376	. . . 4 ⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∩ (V ∖ B))
4		un0 3576	. . . 4 ⊢ (((A ∖ C) ∩ (V ∖ B)) ∪ ∅) = ((A ∖ C) ∩ (V ∖ B))
5	3, 4	eqtr4i 2376	. . 3 ⊢ ((A ∖ B) ∖ C) = (((A ∖ C) ∩ (V ∖ B)) ∪ ∅)
6		indi 3502	. . . 4 ⊢ ((A ∖ C) ∩ ((V ∖ B) ∪ C)) = (((A ∖ C) ∩ (V ∖ B)) ∪ ((A ∖ C) ∩ C))
7		disjdif 3623	. . . . . 6 ⊢ (C ∩ (A ∖ C)) = ∅
8		incom 3449	. . . . . 6 ⊢ (C ∩ (A ∖ C)) = ((A ∖ C) ∩ C)
9	7, 8	eqtr3i 2375	. . . . 5 ⊢ ∅ = ((A ∖ C) ∩ C)
10	9	uneq2i 3416	. . . 4 ⊢ (((A ∖ C) ∩ (V ∖ B)) ∪ ∅) = (((A ∖ C) ∩ (V ∖ B)) ∪ ((A ∖ C) ∩ C))
11	6, 10	eqtr4i 2376	. . 3 ⊢ ((A ∖ C) ∩ ((V ∖ B) ∪ C)) = (((A ∖ C) ∩ (V ∖ B)) ∪ ∅)
12	5, 11	eqtr4i 2376	. 2 ⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∩ ((V ∖ B) ∪ C))
13		ddif 3399	. . . . 5 ⊢ (V ∖ (V ∖ C)) = C
14	13	uneq2i 3416	. . . 4 ⊢ ((V ∖ B) ∪ (V ∖ (V ∖ C))) = ((V ∖ B) ∪ C)
15		indm 3514	. . . . 5 ⊢ (V ∖ (B ∩ (V ∖ C))) = ((V ∖ B) ∪ (V ∖ (V ∖ C)))
16		invdif 3497	. . . . . 6 ⊢ (B ∩ (V ∖ C)) = (B ∖ C)
17	16	difeq2i 3383	. . . . 5 ⊢ (V ∖ (B ∩ (V ∖ C))) = (V ∖ (B ∖ C))
18	15, 17	eqtr3i 2375	. . . 4 ⊢ ((V ∖ B) ∪ (V ∖ (V ∖ C))) = (V ∖ (B ∖ C))
19	14, 18	eqtr3i 2375	. . 3 ⊢ ((V ∖ B) ∪ C) = (V ∖ (B ∖ C))
20	19	ineq2i 3455	. 2 ⊢ ((A ∖ C) ∩ ((V ∖ B) ∪ C)) = ((A ∖ C) ∩ (V ∖ (B ∖ C)))
21		invdif 3497	. 2 ⊢ ((A ∖ C) ∩ (V ∖ (B ∖ C))) = ((A ∖ C) ∖ (B ∖ C))
22	12, 20, 21	3eqtri 2377	1 ⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∖ (B ∖ C))

Syntax hints:   = wceq 1642  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)