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Theorem topnex 23019
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7778; an alternate proof uses indiscrete topologies (see indistop 23025) and the analogue of pwnex 7778 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7776). (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.
StepHypRef Expression
1 pwnex 7778 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 3046 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 distop 23018 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
43elv 3483 . . . . . . 7 𝒫 𝑥 ∈ Top
5 eleq1 2827 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
64, 5mpbiri 258 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
76exlimiv 1928 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87abssi 4080 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
9 ssexg 5329 . . . 4 (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
108, 9mpan 690 . . 3 (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
112, 10mto 197 . 2 ¬ Top ∈ V
1211nelir 3047 1 Top ∉ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wnel 3044  Vcvv 3478  wss 3963  𝒫 cpw 4605  Topctop 22915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-iun 4998  df-top 22916
This theorem is referenced by: (None)
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