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Theorem sssn 3864

Description: The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)

**Assertion**
Ref	Expression
sssn	$\/$

Proof of Theorem sssn Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 3560	. . . . . . 7
2		ssel 3267	. . . . . . . . . . 11
3		elsni 3757	. . . . . . . . . . 11
4	2, 3	syl6 29	. . . . . . . . . 10
5		eleq1 2413	. . . . . . . . . 10
6	4, 5	syl6 29	. . . . . . . . 9
7	6	ibd 234	. . . . . . . 8
8	7	exlimdv 1636	. . . . . . 7
9	1, 8	syl5bi 208	. . . . . 6
10		snssi 3852	. . . . . 6
11	9, 10	syl6 29	. . . . 5
12	11	anc2li 540	. . . 4 $/\$
13		eqss 3287	. . . 4 $/\$
14	12, 13	syl6ibr 218	. . 3
15	14	orrd 367	. 2 $\/$
16		0ss 3579	. . . 4
17		sseq1 3292	. . . 4
18	16, 17	mpbiri 224	. . 3
19		eqimss 3323	. . 3
20	18, 19	jaoi 368	. 2 $\/$
21	15, 20	impbii 180	1 $\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wb 176 $\/$ wo 357 $/\$ wa 358

wex 1541

wceq 1642

wcel 1710

wss 3257

c0 3550

csn 3737

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741

This theorem is referenced by: eqsn 3867 snsssn 3873 pwsn 3881 unsneqsn 3887 foconst 5280