difdifdir - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > difdifdir		Structured version Visualization version Unicode version

Theorem difdifdir 4056

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 3891	. . . . 5 $\$ $\$ $\$ $\$
2		invdif 3868	. . . . 5 $\$ $\$ $\$ $\$
3	1, 2	eqtr4i 2647	. . . 4 $\$ $\$ $\$ $\$
4		un0 3967	. . . 4 $\$ $\$ $\$ $\$
5	3, 4	eqtr4i 2647	. . 3 $\$ $\$ $\$ $\$
6		indi 3873	. . . 4 $\$ $\$ $\$ $\$ $\$
7		disjdif 4040	. . . . . 6 $\$
8		incom 3805	. . . . . 6 $\$ $\$
9	7, 8	eqtr3i 2646	. . . . 5 $\$
10	9	uneq2i 3764	. . . 4 $\$ $\$ $\$ $\$ $\$
11	6, 10	eqtr4i 2647	. . 3 $\$ $\$ $\$ $\$
12	5, 11	eqtr4i 2647	. 2 $\$ $\$ $\$ $\$
13		ddif 3742	. . . . 5 $\$ $\$
14	13	uneq2i 3764	. . . 4 $\$ $\$ $\$ $\$
15		indm 3886	. . . . 5 $\$ $\$ $\$ $\$ $\$
16		invdif 3868	. . . . . 6 $\$ $\$
17	16	difeq2i 3725	. . . . 5 $\$ $\$ $\$ $\$
18	15, 17	eqtr3i 2646	. . . 4 $\$ $\$ $\$ $\$ $\$
19	14, 18	eqtr3i 2646	. . 3 $\$ $\$ $\$
20	19	ineq2i 3811	. 2 $\$ $\$ $\$ $\$ $\$
21		invdif 3868	. 2 $\$ $\$ $\$ $\$ $\$ $\$
22	12, 20, 21	3eqtri 2648	1 $\$ $\$ $\$ $\$ $\$

Colors of variables: wff setvar class

Syntax hints:

wceq 1483

cvv 3200 $\$ cdif 3571

cun 3572

cin 3573

c0 3915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916

This theorem is referenced by: (None)