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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 3086 | . . . . . . . . . 10 | |
2 | elsni 3540 | . . . . . . . . . 10 | |
3 | 1, 2 | syl6 33 | . . . . . . . . 9 |
4 | eleq1 2200 | . . . . . . . . 9 | |
5 | 3, 4 | syl6 33 | . . . . . . . 8 |
6 | 5 | ibd 177 | . . . . . . 7 |
7 | 6 | exlimdv 1791 | . . . . . 6 |
8 | snssi 3659 | . . . . . 6 | |
9 | 7, 8 | syl6 33 | . . . . 5 |
10 | 9 | anc2li 327 | . . . 4 |
11 | eqss 3107 | . . . 4 | |
12 | 10, 11 | syl6ibr 161 | . . 3 |
13 | 12 | com12 30 | . 2 |
14 | eqimss 3146 | . 2 | |
15 | 13, 14 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wss 3066 csn 3522 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-sn 3528 |
This theorem is referenced by: eqsnm 3677 exmid01 4116 exmidn0m 4119 exmidsssn 4120 exmidomni 7007 exmidunben 11928 exmidsbthrlem 13206 sbthomlem 13209 |
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