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Theorem topnex 23003
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7779; an alternate proof uses indiscrete topologies (see indistop 23009) and the analogue of pwnex 7779 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7777). (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 7779 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 3048 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 distop 23002 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
43elv 3485 . . . . . . 7 𝒫 𝑥 ∈ Top
5 eleq1 2829 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
64, 5mpbiri 258 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
76exlimiv 1930 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87abssi 4070 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
9 ssexg 5323 . . . 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 3049 1 Top ∉ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wnel 3046  Vcvv 3480  wss 3951  𝒫 cpw 4600  Topctop 22899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-un 3956  df-in 3958  df-ss 3968  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908  df-iun 4993  df-top 22900
This theorem is referenced by: (None)
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