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Theorem difdifdirss 3577

Description: Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

**Assertion**
Ref	Expression
difdifdirss	$\$ $\$ $\$ $\$ $\$

**Proof of Theorem difdifdirss**
Step	Hyp	Ref	Expression
1		dif32 3468	. . . . 5 $\$ $\$ $\$ $\$
2		invdif 3447	. . . . 5 $\$ $\$ $\$ $\$
3	1, 2	eqtr4i 2253	. . . 4 $\$ $\$ $\$ $\$
4		un0 3526	. . . 4 $\$ $\$ $\$ $\$
5	3, 4	eqtr4i 2253	. . 3 $\$ $\$ $\$ $\$
6		indi 3452	. . . 4 $\$ $\$ $\$ $\$ $\$
7		disjdif 3565	. . . . . 6 $\$
8		incom 3397	. . . . . 6 $\$ $\$
9	7, 8	eqtr3i 2252	. . . . 5 $\$
10	9	uneq2i 3356	. . . 4 $\$ $\$ $\$ $\$ $\$
11	6, 10	eqtr4i 2253	. . 3 $\$ $\$ $\$ $\$
12	5, 11	eqtr4i 2253	. 2 $\$ $\$ $\$ $\$
13		ddifss 3443	. . . . . 6 $\$ $\$
14		unss2 3376	. . . . . 6 $\$ $\$ $\$ $\$ $\$ $\$
15	13, 14	ax-mp 5	. . . . 5 $\$ $\$ $\$ $\$
16		indmss 3464	. . . . . 6 $\$ $\$ $\$ $\$ $\$
17		invdif 3447	. . . . . . 7 $\$ $\$
18	17	difeq2i 3320	. . . . . 6 $\$ $\$ $\$ $\$
19	16, 18	sseqtri 3259	. . . . 5 $\$ $\$ $\$ $\$ $\$
20	15, 19	sstri 3234	. . . 4 $\$ $\$ $\$
21		sslin 3431	. . . 4 $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$
22	20, 21	ax-mp 5	. . 3 $\$ $\$ $\$ $\$ $\$
23		invdif 3447	. . 3 $\$ $\$ $\$ $\$ $\$ $\$
24	22, 23	sseqtri 3259	. 2 $\$ $\$ $\$ $\$ $\$
25	12, 24	eqsstri 3257	1 $\$ $\$ $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

cvv 2800 $\$ cdif 3195

cun 3196

cin 3197

wss 3198

c0 3492

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-ext 2211

This theorem depends on definitions: df-bi 117 df-tru 1398 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rab 2517 df-v 2802 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493

This theorem is referenced by: (None)

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