Theorem topnex 14500

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4495. (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 4495	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 2472	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		vex 2774	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		distop 14499	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
6		eleq1 2267	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
7	5, 6	mpbiri 168	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	exlimiv 1620	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
9	8	abssi 3267	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10		ssexg 4182	. . . 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 663	. 2 ⊢ ¬ Top ∈ V
13	12	nelir 2473	1 ⊢ Top ∉ V

Syntax hints:   = wceq 1372  ∃wex 1514   ∈ wcel 2175  {cab 2190   ∉ wnel 2470  Vcvv 2771   ⊆ wss 3165  𝒫 cpw 3615  Topctop 14411

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-nel 2471  df-ral 2488  df-rex 2489  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-uni 3850  df-iun 3928  df-top 14412

This theorem is referenced by: (None)