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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 2235 | . . . . . . 7 | |
2 | 1 | anidms 395 | . . . . . 6 |
3 | 2 | notbid 662 | . . . . 5 |
4 | df-nel 2436 | . . . . . . 7 | |
5 | eleq12 2235 | . . . . . . . . 9 | |
6 | 5 | anidms 395 | . . . . . . . 8 |
7 | 6 | notbid 662 | . . . . . . 7 |
8 | 4, 7 | syl5bb 191 | . . . . . 6 |
9 | 8 | cbvrabv 2729 | . . . . 5 |
10 | 3, 9 | elrab2 2889 | . . . 4 |
11 | pclem6 1369 | . . . 4 | |
12 | 10, 11 | ax-mp 5 | . . 3 |
13 | ssel 3141 | . . 3 | |
14 | 12, 13 | mtoi 659 | . 2 |
15 | ssrab2 3232 | . . 3 | |
16 | elpw2g 4140 | . . 3 | |
17 | 15, 16 | mpbiri 167 | . 2 |
18 | 14, 17 | nsyl3 621 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wnel 2435 crab 2452 wss 3121 cpw 3564 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4105 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-nel 2436 df-rab 2457 df-v 2732 df-in 3127 df-ss 3134 df-pw 3566 |
This theorem is referenced by: pwne 4144 pwuninel2 6258 |
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