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Theorem pwne 4978
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4583. (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 4977 . 2 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
2 eqimss 3796 . . 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 2137  wne 2930  wss 3713  𝒫 cpw 4300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-rab 3057  df-v 3340  df-in 3720  df-ss 3727  df-pw 4302
This theorem is referenced by:  pnfnemnf  10284
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