difdifdirss - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > difdifdirss		Unicode version

Theorem difdifdirss 3478

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 3370	. . . . 5 $\$ $\$ $\$ $\$
2		invdif 3349	. . . . 5 $\$ $\$ $\$ $\$
3	1, 2	eqtr4i 2181	. . . 4 $\$ $\$ $\$ $\$
4		un0 3427	. . . 4 $\$ $\$ $\$ $\$
5	3, 4	eqtr4i 2181	. . 3 $\$ $\$ $\$ $\$
6		indi 3354	. . . 4 $\$ $\$ $\$ $\$ $\$
7		disjdif 3466	. . . . . 6 $\$
8		incom 3299	. . . . . 6 $\$ $\$
9	7, 8	eqtr3i 2180	. . . . 5 $\$
10	9	uneq2i 3258	. . . 4 $\$ $\$ $\$ $\$ $\$
11	6, 10	eqtr4i 2181	. . 3 $\$ $\$ $\$ $\$
12	5, 11	eqtr4i 2181	. 2 $\$ $\$ $\$ $\$
13		ddifss 3345	. . . . . 6 $\$ $\$
14		unss2 3278	. . . . . 6 $\$ $\$ $\$ $\$ $\$ $\$
15	13, 14	ax-mp 5	. . . . 5 $\$ $\$ $\$ $\$
16		indmss 3366	. . . . . 6 $\$ $\$ $\$ $\$ $\$
17		invdif 3349	. . . . . . 7 $\$ $\$
18	17	difeq2i 3222	. . . . . 6 $\$ $\$ $\$ $\$
19	16, 18	sseqtri 3162	. . . . 5 $\$ $\$ $\$ $\$ $\$
20	15, 19	sstri 3137	. . . 4 $\$ $\$ $\$
21		sslin 3333	. . . 4 $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$
22	20, 21	ax-mp 5	. . 3 $\$ $\$ $\$ $\$ $\$
23		invdif 3349	. . 3 $\$ $\$ $\$ $\$ $\$ $\$
24	22, 23	sseqtri 3162	. 2 $\$ $\$ $\$ $\$ $\$
25	12, 24	eqsstri 3160	1 $\$ $\$ $\$ $\$ $\$

Colors of variables: wff set class

Syntax hints:

cvv 2712 $\$ cdif 3099

cun 3100

cin 3101

wss 3102

c0 3394

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139

This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395

This theorem is referenced by: (None)

W3C validator