Theorem pwnex 7756

Description: The class of all power sets is a proper class. See also snnex 7755. (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 7754	. . 3 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
2		df-nel 3064	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
3	1, 2	sylibr 237	. 2 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V)
4		vpwex 5347	. . 3 ⊢ 𝒫 𝑦 ∈ V
5		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
6	5	pwid 4584	. . 3 ⊢ 𝑦 ∈ 𝒫 𝑦
7	4, 6	pm3.2i 475	. 2 ⊢ (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
8	3, 7	mpg 1826	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 400  ∀wal 1567   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740   ∉ wnel 3063  Vcvv 3454  𝒫 cpw 4561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5256  ax-pow 5335  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-in 3911  df-ss 3921  df-pw 4563  df-sn 4589  df-uni 4872  df-iun 4957

This theorem is used by:  topnex  23164