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Mirrors > Home > ILE Home > Th. List > pwne | Unicode version |
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3806. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
pwne |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnss 4156 |
. 2
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2 | eqimss 3209 |
. . 3
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3 | 2 | necon3bi 2397 |
. 2
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4 | 1, 3 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4118 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-rab 2464 df-v 2739 df-in 3135 df-ss 3142 df-pw 3576 |
This theorem is referenced by: pnfnemnf 7989 |
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