Theorem difdifdir 4446

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 4246	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
2		invdif 4223	. . . . 5 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
3	1, 2	eqtr4i 2786	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
4		un0 4343	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
5	3, 4	eqtr4i 2786	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6		indi 4228	. . . 4 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
7		disjdif 4425	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅
8		incom 4105	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐶)
9	7, 8	eqtr3i 2785	. . . . 5 ⊢ ∅ = ((𝐴 ∖ 𝐶) ∩ 𝐶)
10	9	uneq2i 4163	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
11	6, 10	eqtr4i 2786	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
12	5, 11	eqtr4i 2786	. 2 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13		ddif 4076	. . . . 5 ⊢ (V ∖ (V ∖ 𝐶)) = 𝐶
14	13	uneq2i 4163	. . . 4 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ 𝐶)
15		indm 4242	. . . . 5 ⊢ (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶)))
16		invdif 4223	. . . . . 6 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
17	16	difeq2i 4059	. . . . 5 ⊢ (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵 ∖ 𝐶))
18	15, 17	eqtr3i 2785	. . . 4 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = (V ∖ (𝐵 ∖ 𝐶))
19	14, 18	eqtr3i 2785	. . 3 ⊢ ((V ∖ 𝐵) ∪ 𝐶) = (V ∖ (𝐵 ∖ 𝐶))
20	19	ineq2i 4113	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶)))
21		invdif 4223	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
22	12, 20, 21	3eqtri 2787	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3450   ∖ cdif 4044   ∩ cin 4088   ∪ cun 4148  ∅c0 4277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278

This theorem is used by: (None)