Theorem difdifdir 4445

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 4254	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
2		invdif 4231	. . . . 5 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
3	1, 2	eqtr4i 2788	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
4		un0 4348	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
5	3, 4	eqtr4i 2788	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6		indi 4236	. . . 4 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
7		disjdif 4426	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅
8		incom 4161	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐶)
9	7, 8	eqtr3i 2787	. . . . 5 ⊢ ∅ = ((𝐴 ∖ 𝐶) ∩ 𝐶)
10	9	uneq2i 4118	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
11	6, 10	eqtr4i 2788	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
12	5, 11	eqtr4i 2788	. 2 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13		ddif 4094	. . . . 5 ⊢ (V ∖ (V ∖ 𝐶)) = 𝐶
14	13	uneq2i 4118	. . . 4 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ 𝐶)
15		indm 4250	. . . . 5 ⊢ (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶)))
16		invdif 4231	. . . . . 6 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
17	16	difeq2i 4077	. . . . 5 ⊢ (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵 ∖ 𝐶))
18	15, 17	eqtr3i 2787	. . . 4 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = (V ∖ (𝐵 ∖ 𝐶))
19	14, 18	eqtr3i 2787	. . 3 ⊢ ((V ∖ 𝐵) ∪ 𝐶) = (V ∖ (𝐵 ∖ 𝐶))
20	19	ineq2i 4169	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶)))
21		invdif 4231	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
22	12, 20, 21	3eqtri 2789	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))

Syntax hints:   = wceq 1560  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286

This theorem is referenced by: (None)