Theorem topnex 15077

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4575. (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 4575	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 2511	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		vex 2818	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		distop 15076	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
6		eleq1 2297	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
7	5, 6	mpbiri 168	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	exlimiv 1647	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
9	8	abssi 3317	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10		ssexg 4254	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
11	9, 10	mpan 424	. . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
12	2, 11	mto 668	. 2 ⊢ ¬ Top ∈ V
13	12	nelir 2512	1 ⊢ Top ∉ V

Syntax hints:   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {cab 2220   ∉ wnel 2509  Vcvv 2815   ⊆ wss 3214  𝒫 cpw 3674  Topctop 14988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-nel 2510  df-ral 2527  df-rex 2528  df-v 2817  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-uni 3920  df-iun 3998  df-top 14989

This theorem is referenced by: (None)