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Mirrors > Home > ILE Home > Th. List > mss | Unicode version |
Description: An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.) |
Ref | Expression |
---|---|
mss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2684 | . . . . 5 | |
2 | 1 | snss 3644 | . . . 4 |
3 | 1 | snm 3638 | . . . . 5 |
4 | 1 | snex 4104 | . . . . . 6 |
5 | sseq1 3115 | . . . . . . 7 | |
6 | eleq2 2201 | . . . . . . . 8 | |
7 | 6 | exbidv 1797 | . . . . . . 7 |
8 | 5, 7 | anbi12d 464 | . . . . . 6 |
9 | 4, 8 | spcev 2775 | . . . . 5 |
10 | 3, 9 | mpan2 421 | . . . 4 |
11 | 2, 10 | sylbi 120 | . . 3 |
12 | 11 | exlimiv 1577 | . 2 |
13 | elequ1 1690 | . . . . 5 | |
14 | 13 | cbvexv 1890 | . . . 4 |
15 | 14 | anbi2i 452 | . . 3 |
16 | 15 | exbii 1584 | . 2 |
17 | 12, 16 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 |
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-pw 3507 df-sn 3528 |
This theorem is referenced by: (None) |
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