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Theorem pwne 3904
 Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3570. (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 3903 . 2 (A 𝑉 → ¬ 𝒫 AA)
2 eqimss 2991 . . 3 (𝒫 A = A → 𝒫 AA)
32necon3bi 2249 . 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 1390   ≠ wne 2201   ⊆ wss 2911  𝒫 cpw 3351 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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353 This theorem is referenced by:  pnfnemnf  8447
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