Theorem pwnex 7794

Description: The class of all power sets is a proper class. See also snnex 7793. (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 7792	. . 3 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
2		df-nel 3053	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
3	1, 2	sylibr 234	. 2 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V)
4		vpwex 5395	. . 3 ⊢ 𝒫 𝑦 ∈ V
5		vex 3492	. . . 4 ⊢ 𝑦 ∈ V
6	5	pwid 4644	. . 3 ⊢ 𝑦 ∈ 𝒫 𝑦
7	4, 6	pm3.2i 470	. 2 ⊢ (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
8	3, 7	mpg 1795	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717   ∉ wnel 3052  Vcvv 3488  𝒫 cpw 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-pow 5383  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-pw 4624  df-sn 4649  df-uni 4932  df-iun 5017

This theorem is referenced by:  topnex  23024