Theorem topnex 14754

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4539. (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 4539	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 2497	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		vex 2802	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		distop 14753	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
6		eleq1 2292	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
7	5, 6	mpbiri 168	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	exlimiv 1644	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
9	8	abssi 3299	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10		ssexg 4222	. . . 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 666	. 2 ⊢ ¬ Top ∈ V
13	12	nelir 2498	1 ⊢ Top ∉ V

Syntax hints:   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215   ∉ wnel 2495  Vcvv 2799   ⊆ wss 3197  𝒫 cpw 3649  Topctop 14665

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-nel 2496  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-uni 3888  df-iun 3966  df-top 14666

This theorem is referenced by: (None)