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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 2229 | . . . . . . 7 | |
2 | 1 | anidms 395 | . . . . . 6 |
3 | 2 | notbid 657 | . . . . 5 |
4 | df-nel 2430 | . . . . . . 7 | |
5 | eleq12 2229 | . . . . . . . . 9 | |
6 | 5 | anidms 395 | . . . . . . . 8 |
7 | 6 | notbid 657 | . . . . . . 7 |
8 | 4, 7 | syl5bb 191 | . . . . . 6 |
9 | 8 | cbvrabv 2720 | . . . . 5 |
10 | 3, 9 | elrab2 2880 | . . . 4 |
11 | pclem6 1363 | . . . 4 | |
12 | 10, 11 | ax-mp 5 | . . 3 |
13 | ssel 3131 | . . 3 | |
14 | 12, 13 | mtoi 654 | . 2 |
15 | ssrab2 3222 | . . 3 | |
16 | elpw2g 4129 | . . 3 | |
17 | 15, 16 | mpbiri 167 | . 2 |
18 | 14, 17 | nsyl3 616 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wnel 2429 crab 2446 wss 3111 cpw 3553 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-nel 2430 df-rab 2451 df-v 2723 df-in 3117 df-ss 3124 df-pw 3555 |
This theorem is referenced by: pwne 4133 pwuninel2 6241 |
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