Theorem pwnex 7742

Description: The class of all power sets is a proper class. See also snnex 7741. (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 7740	. . 3 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
2		df-nel 3063	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
3	1, 2	sylibr 236	. 2 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V)
4		vpwex 5335	. . 3 ⊢ 𝒫 𝑦 ∈ V
5		vex 3459	. . . 4 ⊢ 𝑦 ∈ V
6	5	pwid 4579	. . 3 ⊢ 𝑦 ∈ 𝒫 𝑦
7	4, 6	pm3.2i 474	. 2 ⊢ (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
8	3, 7	mpg 1818	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

Syntax hints:  ¬ wn 3   ∧ wa 399  ∀wal 1559   = wceq 1561  ∃wex 1800   ∈ wcel 2143  {cab 2741   ∉ wnel 3062  Vcvv 3455  𝒫 cpw 4556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5247  ax-pow 5323  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1101  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-in 3912  df-ss 3922  df-pw 4558  df-sn 4584  df-uni 4867  df-iun 4952

This theorem is referenced by:  topnex  23063