Theorem bj-pwnex 32246

Description: The class of all power sets is a proper class. See also snnex 6862. (Contributed by BJ, 2-May-2021.)

Assertion

Ref	Expression
bj-pwnex	⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-pwnex

Step	Hyp	Ref	Expression
1		bj-xnex 32245	. 2 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V)
2		vpwex 4775	. . 3 ⊢ 𝒫 𝑦 ∈ V
3		vex 3176	. . . 4 ⊢ 𝑦 ∈ V
4	3	pwid 4122	. . 3 ⊢ 𝑦 ∈ 𝒫 𝑦
5	2, 4	pm3.2i 470	. 2 ⊢ (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
6	1, 5	mpg 1715	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ∉ wnel 2781  Vcvv 3173  𝒫 cpw 4108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-pow 4769  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373

This theorem is referenced by:  bj-topnex  32247