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Theorem pwne 3887
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3553. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
pwne (A 𝑉 → 𝒫 AA)

Proof of Theorem pwne
StepHypRef Expression
1 pwnss 3886 . 2 (A 𝑉 → ¬ 𝒫 AA)
2 eqimss 2974 . . 3 (𝒫 A = A → 𝒫 AA)
32necon3bi 2233 . 2 (¬ 𝒫 AA → 𝒫 AA)
41, 3syl 14 1 (A 𝑉 → 𝒫 AA)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wcel 1374  wne 2186  wss 2894  𝒫 cpw 3334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-nel 2189  df-rab 2293  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336
This theorem is referenced by: (None)
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