Theorem difdifdir 1765

Description: Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16.

Assertion

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

Proof of Theorem difdifdir

Step	Hyp	Ref	Expression
1		difdisj 1758	. . . . 5 ⊢ (C ∩ (A ∖ C)) = ∅
2		incom 1636	. . . . 5 ⊢ (C ∩ (A ∖ C)) = ((A ∖ C) ∩ C)
3	1, 2	eqtr3 1121	. . . 4 ⊢ ∅ = ((A ∖ C) ∩ C)
4	3	uneq2i 1608	. . 3 ⊢ (((A ∖ C) ∩ (V ∖ B)) ∪ ∅) = (((A ∖ C) ∩ (V ∖ B)) ∪ ((A ∖ C) ∩ C))
5		invdif 1674	. . . 4 ⊢ ((A ∖ C) ∩ (V ∖ B)) = ((A ∖ C) ∖ B)
6		un0 1721	. . . 4 ⊢ (((A ∖ C) ∩ (V ∖ B)) ∪ ∅) = ((A ∖ C) ∩ (V ∖ B))
7		dif23 1688	. . . 4 ⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∖ B)
8	5, 6, 7	3eqtr4r 1127	. . 3 ⊢ ((A ∖ B) ∖ C) = (((A ∖ C) ∩ (V ∖ B)) ∪ ∅)
9		indi 1676	. . 3 ⊢ ((A ∖ C) ∩ ((V ∖ B) ∪ C)) = (((A ∖ C) ∩ (V ∖ B)) ∪ ((A ∖ C) ∩ C))
10	4, 8, 9	3eqtr4 1126	. 2 ⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∩ ((V ∖ B) ∪ C))
11		indm 1686	. . . 4 ⊢ (V ∖ (B ∩ (V ∖ C))) = ((V ∖ B) ∪ (V ∖ (V ∖ C)))
12		invdif 1674	. . . . 5 ⊢ (B ∩ (V ∖ C)) = (B ∖ C)
13	12	difeq2i 1585	. . . 4 ⊢ (V ∖ (B ∩ (V ∖ C))) = (V ∖ (B ∖ C))
14		ddif 1597	. . . . 5 ⊢ (V ∖ (V ∖ C)) = C
15	14	uneq2i 1608	. . . 4 ⊢ ((V ∖ B) ∪ (V ∖ (V ∖ C))) = ((V ∖ B) ∪ C)
16	11, 13, 15	3eqtr3r 1125	. . 3 ⊢ ((V ∖ B) ∪ C) = (V ∖ (B ∖ C))
17	16	ineq2i 1642	. 2 ⊢ ((A ∖ C) ∩ ((V ∖ B) ∪ C)) = ((A ∖ C) ∩ (V ∖ (B ∖ C)))
18		invdif 1674	. 2 ⊢ ((A ∖ C) ∩ (V ∖ (B ∖ C))) = ((A ∖ C) ∖ (B ∖ C))
19	10, 17, 18	3eqtr 1123	1 ⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∖ (B ∖ C))

Syntax hints:   = wceq 1091  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486  ∅c0 1707

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708

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