Theorem topnex 22984

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7756; an alternate proof uses indiscrete topologies (see indistop 22990) and the analogue of pwnex 7756 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7754). (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 7756	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 3038	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		distop 22983	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
4	3	elv 3468	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
5		eleq1 2814	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
6	4, 5	mpbiri 257	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
7	6	exlimiv 1926	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	abssi 4063	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
9		ssexg 5318	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
10	8, 9	mpan 688	. . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
11	2, 10	mto 196	. 2 ⊢ ¬ Top ∈ V
12	11	nelir 3039	1 ⊢ Top ∉ V

Syntax hints:   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2703   ∉ wnel 3036  Vcvv 3462   ⊆ wss 3946  𝒫 cpw 4597  Topctop 22880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2697  ax-sep 5294  ax-pow 5359  ax-pr 5423  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-un 3951  df-in 3953  df-ss 3963  df-pw 4599  df-sn 4624  df-pr 4626  df-uni 4906  df-iun 4995  df-top 22881

This theorem is referenced by: (None)