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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 2102 | . . . . . . 7 | |
2 | 1 | anidms 377 | . . . . . 6 |
3 | 2 | notbid 592 | . . . . 5 |
4 | df-nel 2207 | . . . . . . 7 | |
5 | eleq12 2102 | . . . . . . . . 9 | |
6 | 5 | anidms 377 | . . . . . . . 8 |
7 | 6 | notbid 592 | . . . . . . 7 |
8 | 4, 7 | syl5bb 181 | . . . . . 6 |
9 | 8 | cbvrabv 2556 | . . . . 5 |
10 | 3, 9 | elrab2 2700 | . . . 4 |
11 | pclem6 1265 | . . . 4 | |
12 | 10, 11 | ax-mp 7 | . . 3 |
13 | ssel 2939 | . . 3 | |
14 | 12, 13 | mtoi 590 | . 2 |
15 | ssrab2 3025 | . . 3 | |
16 | elpw2g 3910 | . . 3 | |
17 | 15, 16 | mpbiri 157 | . 2 |
18 | 14, 17 | nsyl3 556 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wnel 2205 crab 2310 wss 2917 cpw 3359 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-nel 2207 df-rab 2315 df-v 2559 df-in 2924 df-ss 2931 df-pw 3361 |
This theorem is referenced by: pwne 3913 pwuninel2 5897 |
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