Theorem topnex 12255

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4370. (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 4370	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 2405	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		vex 2689	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		distop 12254	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
6		eleq1 2202	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
7	5, 6	mpbiri 167	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	exlimiv 1577	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
9	8	abssi 3172	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10		ssexg 4067	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
11	9, 10	mpan 420	. . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
12	2, 11	mto 651	. 2 ⊢ ¬ Top ∈ V
13	12	nelir 2406	1 ⊢ Top ∉ V

Syntax hints:   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2125   ∉ wnel 2403  Vcvv 2686   ⊆ wss 3071  𝒫 cpw 3510  Topctop 12164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-nel 2404  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-uni 3737  df-iun 3815  df-top 12165

This theorem is referenced by: (None)