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Theorem pwnex 7779
Description: The class of all power sets is a proper class. See also snnex 7778. (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 7777 . . 3 (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
2 df-nel 3047 . . 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 5377 . . 3 𝒫 𝑦 ∈ V
5 vex 3484 . . . 4 𝑦 ∈ V
65pwid 4622 . . 3 𝑦 ∈ 𝒫 𝑦
74, 6pm3.2i 470 . 2 (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
83, 7mpg 1797 1 {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wnel 3046  Vcvv 3480  𝒫 cpw 4600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-pow 5365  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602  df-sn 4627  df-uni 4908  df-iun 4993
This theorem is referenced by:  topnex  23003
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