Theorem difdifdir 3915

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

Assertion

Ref	Expression
difdifdir

Proof of Theorem difdifdir

Step	Hyp	Ref	Expression
1		dif32 3760	. . . . 5
2		invdif 3738	. . . . 5
3	1, 2	eqtr4i 2489	. . . 4
4		un0 3810	. . . 4
5	3, 4	eqtr4i 2489	. . 3
6		indi 3743	. . . 4
7		disjdif 3900	. . . . . 6
8		incom 3690	. . . . . 6
9	7, 8	eqtr3i 2488	. . . . 5
10	9	uneq2i 3654	. . . 4
11	6, 10	eqtr4i 2489	. . 3
12	5, 11	eqtr4i 2489	. 2
13		ddif 3635	. . . . 5
14	13	uneq2i 3654	. . . 4
15		indm 3756	. . . . 5
16		invdif 3738	. . . . . 6
17	16	difeq2i 3618	. . . . 5
18	15, 17	eqtr3i 2488	. . . 4
19	14, 18	eqtr3i 2488	. . 3
20	19	ineq2i 3696	. 2
21		invdif 3738	. 2
22	12, 20, 21	3eqtri 2490	1

Syntax hints:  =wceq 1395  cvv 3109  \cdif 3472  u.cun 3473  i^icin 3474  c0 3784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785