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Theorem pwne 3940
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3606. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
pwne (𝐴𝑉 → 𝒫 𝐴𝐴)

Proof of Theorem pwne
StepHypRef Expression
1 pwnss 3939 . 2 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
2 eqimss 3024 . . 3 (𝒫 𝐴 = 𝐴 → 𝒫 𝐴𝐴)
32necon3bi 2270 . 2 (¬ 𝒫 𝐴𝐴 → 𝒫 𝐴𝐴)
41, 3syl 14 1 (𝐴𝑉 → 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1409  wne 2220  wss 2944  𝒫 cpw 3386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-rab 2332  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388
This theorem is referenced by:  pnfnemnf  8797
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