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Theorem pwnex 7671
Description: The class of all power sets is a proper class. See also snnex 7670. (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 7669 . . 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 233 . 2 (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V)
4 vpwex 5320 . . 3 𝒫 𝑦 ∈ V
5 vex 3445 . . . 4 𝑦 ∈ V
65pwid 4569 . . 3 𝑦 ∈ 𝒫 𝑦
74, 6pm3.2i 471 . 2 (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
83, 7mpg 1798 1 {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wal 1538   = wceq 1540  wex 1780  wcel 2105  {cab 2713  wnel 3046  Vcvv 3441  𝒫 cpw 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2707  ax-sep 5243  ax-pow 5308  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-in 3905  df-ss 3915  df-pw 4549  df-sn 4574  df-uni 4853  df-iun 4943
This theorem is referenced by:  topnex  22252
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