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Theorem pwne 4757
 Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4371. (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 4756 . 2 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
2 eqimss 3620 . . 3 (𝒫 𝐴 = 𝐴 → 𝒫 𝐴𝐴)
32necon3bi 2808 . 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 1977   ≠ wne 2780   ⊆ wss 3540  𝒫 cpw 4108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110 This theorem is referenced by:  pnfnemnf  9973
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