Theorem topnex 23314

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7773; an alternate proof uses indiscrete topologies (see indistop 23320) and the analogue of pwnex 7773 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7771). (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 7773	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 3064	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		distop 23313	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
4	3	elv 3456	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
5		eleq1 2849	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
6	4, 5	mpbiri 261	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
7	6	exlimiv 1963	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	abssi 4016	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
9		ssexg 5281	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
10	8, 9	mpan 703	. . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
11	2, 10	mto 200	. 2 ⊢ ¬ Top ∈ V
12	11	nelir 3065	1 ⊢ Top ∉ V

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739   ∉ wnel 3062  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  Topctop 23211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2733  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-un 3904  df-in 3906  df-ss 3916  df-pw 4559  df-sn 4585  df-pr 4587  df-uni 4868  df-iun 4953  df-top 23212

This theorem is used by: (None)