Theorem difdifdir 4436

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 4266	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
2		invdif 4244	. . . . 5 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
3	1, 2	eqtr4i 2847	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
4		un0 4343	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵))
5	3, 4	eqtr4i 2847	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6		indi 4249	. . . 4 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
7		disjdif 4420	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅
8		incom 4177	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐶)
9	7, 8	eqtr3i 2846	. . . . 5 ⊢ ∅ = ((𝐴 ∖ 𝐶) ∩ 𝐶)
10	9	uneq2i 4135	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴 ∖ 𝐶) ∩ 𝐶))
11	6, 10	eqtr4i 2847	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴 ∖ 𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
12	5, 11	eqtr4i 2847	. 2 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13		ddif 4112	. . . . 5 ⊢ (V ∖ (V ∖ 𝐶)) = 𝐶
14	13	uneq2i 4135	. . . 4 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ 𝐶)
15		indm 4262	. . . . 5 ⊢ (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶)))
16		invdif 4244	. . . . . 6 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
17	16	difeq2i 4095	. . . . 5 ⊢ (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵 ∖ 𝐶))
18	15, 17	eqtr3i 2846	. . . 4 ⊢ ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = (V ∖ (𝐵 ∖ 𝐶))
19	14, 18	eqtr3i 2846	. . 3 ⊢ ((V ∖ 𝐵) ∪ 𝐶) = (V ∖ (𝐵 ∖ 𝐶))
20	19	ineq2i 4185	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶)))
21		invdif 4244	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ (V ∖ (𝐵 ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
22	12, 20, 21	3eqtri 2848	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))

Syntax hints:   = wceq 1533  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ∩ cin 3934  ∅c0 4290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291

This theorem is referenced by: (None)