Theorem pwnex 7600

Description: The class of all power sets is a proper class. See also snnex 7599. (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

Step	Hyp	Ref	Expression
1		abnex 7598	. . 3 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
2		df-nel 3051	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
3	1, 2	sylibr 233	. 2 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V)
4		vpwex 5303	. . 3 ⊢ 𝒫 𝑦 ∈ V
5		vex 3434	. . . 4 ⊢ 𝑦 ∈ V
6	5	pwid 4562	. . 3 ⊢ 𝑦 ∈ 𝒫 𝑦
7	4, 6	pm3.2i 470	. 2 ⊢ (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
8	3, 7	mpg 1803	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1785   ∈ wcel 2109  {cab 2716   ∉ wnel 3050  Vcvv 3430  𝒫 cpw 4538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-pow 5291  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-nel 3051  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-in 3898  df-ss 3908  df-pw 4540  df-sn 4567  df-uni 4845  df-iun 4931

This theorem is referenced by:  topnex  22127