Theorem topnex 23024

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7794; an alternate proof uses indiscrete topologies (see indistop 23030) and the analogue of pwnex 7794 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7792). (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 7794	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 3054	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		distop 23023	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
4	3	elv 3493	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
5		eleq1 2832	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
6	4, 5	mpbiri 258	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
7	6	exlimiv 1929	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	abssi 4093	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
9		ssexg 5341	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
10	8, 9	mpan 689	. . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
11	2, 10	mto 197	. 2 ⊢ ¬ Top ∈ V
12	11	nelir 3055	1 ⊢ Top ∉ V

Syntax hints:   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717   ∉ wnel 3052  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  Topctop 22920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-un 3981  df-in 3983  df-ss 3993  df-pw 4624  df-sn 4649  df-pr 4651  df-uni 4932  df-iun 5017  df-top 22921

This theorem is referenced by: (None)