Theorem topnex 22952

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7714; an alternate proof uses indiscrete topologies (see indistop 22958) and the analogue of pwnex 7714 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7712). (Contributed by BJ, 2-May-2021.)

Assertion

Ref	Expression
topnex	⊢ Top ∉ V

Proof of Theorem topnex

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwnex 7714	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 3039	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		distop 22951	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
4	3	elv 3447	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
5		eleq1 2825	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
6	4, 5	mpbiri 258	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
7	6	exlimiv 1932	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	abssi 4022	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
9		ssexg 5270	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
10	8, 9	mpan 691	. . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
11	2, 10	mto 197	. 2 ⊢ ¬ Top ∈ V
12	11	nelir 3040	1 ⊢ Top ∉ V

Syntax hints:   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ∉ wnel 3037  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556  Topctop 22849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-un 3908  df-in 3910  df-ss 3920  df-pw 4558  df-sn 4583  df-pr 4585  df-uni 4866  df-iun 4950  df-top 22850

This theorem is referenced by: (None)