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Theorem pwne 5353
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4904. (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 5352 . 2 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
2 eqimss 4042 . . 3 (𝒫 𝐴 = 𝐴 → 𝒫 𝐴𝐴)
32necon3bi 2967 . 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 2108  wne 2940  wss 3951  𝒫 cpw 4600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602
This theorem is referenced by:  pnfnemnf  11316
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