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Mirrors > Home > ILE Home > Th. List > sssnm | Unicode version |
Description: The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
sssnm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3122 | . . . . . . . . . 10 | |
2 | elsni 3578 | . . . . . . . . . 10 | |
3 | 1, 2 | syl6 33 | . . . . . . . . 9 |
4 | eleq1 2220 | . . . . . . . . 9 | |
5 | 3, 4 | syl6 33 | . . . . . . . 8 |
6 | 5 | ibd 177 | . . . . . . 7 |
7 | 6 | exlimdv 1799 | . . . . . 6 |
8 | snssi 3700 | . . . . . 6 | |
9 | 7, 8 | syl6 33 | . . . . 5 |
10 | 9 | anc2li 327 | . . . 4 |
11 | eqss 3143 | . . . 4 | |
12 | 10, 11 | syl6ibr 161 | . . 3 |
13 | 12 | com12 30 | . 2 |
14 | eqimss 3182 | . 2 | |
15 | 13, 14 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 wss 3102 csn 3560 |
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-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-v 2714 df-in 3108 df-ss 3115 df-sn 3566 |
This theorem is referenced by: eqsnm 3718 ss1o0el1 4158 exmidn0m 4162 exmidsssn 4163 exmidomni 7085 exmidunben 12166 exmidsbthrlem 13604 sbthomlem 13607 |
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