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Theorem pwne 5352
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4906. (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 5351 . 2 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
2 eqimss 4035 . . 3 (𝒫 𝐴 = 𝐴 → 𝒫 𝐴𝐴)
32necon3bi 2956 . 2 (¬ 𝒫 𝐴𝐴 → 𝒫 𝐴𝐴)
41, 3syl 17 1 (𝐴𝑉 → 𝒫 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2098  wne 2929  wss 3944  𝒫 cpw 4604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-rab 3419  df-v 3463  df-in 3951  df-ss 3961  df-pw 4606
This theorem is referenced by:  pnfnemnf  11301
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