Theorem pwnex 7603

Description: The class of all power sets is a proper class. See also snnex 7602. (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 7601	. . 3 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
2		df-nel 3052	. . 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 5304	. . 3 ⊢ 𝒫 𝑦 ∈ V
5		vex 3435	. . . 4 ⊢ 𝑦 ∈ V
6	5	pwid 4563	. . 3 ⊢ 𝑦 ∈ 𝒫 𝑦
7	4, 6	pm3.2i 471	. 2 ⊢ (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
8	3, 7	mpg 1804	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

Syntax hints:  ¬ wn 3   ∧ wa 396  ∀wal 1540   = wceq 1542  ∃wex 1786   ∈ wcel 2110  {cab 2717   ∉ wnel 3051  Vcvv 3431  𝒫 cpw 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-pow 5292  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-nel 3052  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-in 3899  df-ss 3909  df-pw 4541  df-sn 4568  df-uni 4846  df-iun 4932

This theorem is referenced by:  topnex  22144