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Mirrors > Home > NFE Home > Th. List > sssn | Unicode version |
Description: The subsets of a singleton. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
sssn |
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-mp 5 ax-1 6 ax-2 7 ax-3 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 |
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