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Mirrors > Home > ILE Home > Th. List > pwnss | Unicode version |
Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
pwnss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq12 2202 | . . . . . . 7 | |
2 | 1 | anidms 394 | . . . . . 6 |
3 | 2 | notbid 656 | . . . . 5 |
4 | df-nel 2402 | . . . . . . 7 | |
5 | eleq12 2202 | . . . . . . . . 9 | |
6 | 5 | anidms 394 | . . . . . . . 8 |
7 | 6 | notbid 656 | . . . . . . 7 |
8 | 4, 7 | syl5bb 191 | . . . . . 6 |
9 | 8 | cbvrabv 2680 | . . . . 5 |
10 | 3, 9 | elrab2 2838 | . . . 4 |
11 | pclem6 1352 | . . . 4 | |
12 | 10, 11 | ax-mp 5 | . . 3 |
13 | ssel 3086 | . . 3 | |
14 | 12, 13 | mtoi 653 | . 2 |
15 | ssrab2 3177 | . . 3 | |
16 | elpw2g 4076 | . . 3 | |
17 | 15, 16 | mpbiri 167 | . 2 |
18 | 14, 17 | nsyl3 615 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wnel 2401 crab 2418 wss 3066 cpw 3505 |
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-in1 603 ax-in2 604 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 ax-sep 4041 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-nel 2402 df-rab 2423 df-v 2683 df-in 3072 df-ss 3079 df-pw 3507 |
This theorem is referenced by: pwne 4079 pwuninel2 6172 |
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