Theorem topnex 14322

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4484. (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 4484	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 2464	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		vex 2766	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		distop 14321	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
6		eleq1 2259	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
7	5, 6	mpbiri 168	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	exlimiv 1612	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
9	8	abssi 3258	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10		ssexg 4172	. . . 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 2465	1 ⊢ Top ∉ V

Syntax hints:   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182   ∉ wnel 2462  Vcvv 2763   ⊆ wss 3157  𝒫 cpw 3605  Topctop 14233

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-nel 2463  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-uni 3840  df-iun 3918  df-top 14234

This theorem is referenced by: (None)