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Theorem pwnex 7778
Description: The class of all power sets is a proper class. See also snnex 7777. (Contributed by BJ, 2-May-2021.)
Assertion
Ref Expression
pwnex {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V
Distinct variable group:   𝑥,𝑦

Proof of Theorem pwnex
StepHypRef Expression
1 abnex 7776 . . 3 (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
2 df-nel 3045 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
31, 2sylibr 234 . 2 (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V)
4 vpwex 5383 . . 3 𝒫 𝑦 ∈ V
5 vex 3482 . . . 4 𝑦 ∈ V
65pwid 4627 . . 3 𝑦 ∈ 𝒫 𝑦
74, 6pm3.2i 470 . 2 (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
83, 7mpg 1794 1 {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wnel 3044  Vcvv 3478  𝒫 cpw 4605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-pow 5371  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607  df-sn 4632  df-uni 4913  df-iun 4998
This theorem is referenced by:  topnex  23019
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