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Theorem topnex 21292
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7345; an alternate proof uses indiscrete topologies (see indistop 21298) and the analogue of pwnex 7345 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7343). (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 7345 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 3094 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 distop 21291 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
43elv 3445 . . . . . . 7 𝒫 𝑥 ∈ Top
5 eleq1 2872 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
64, 5mpbiri 259 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
76exlimiv 1912 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87abssi 3973 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
9 ssexg 5125 . . . 4 (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
108, 9mpan 686 . . 3 (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
112, 10mto 198 . 2 ¬ Top ∈ V
1211nelir 3095 1 Top ∉ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1525  wex 1765  wcel 2083  {cab 2777  wnel 3092  Vcvv 3440  wss 3865  𝒫 cpw 4459  Topctop 21189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-nel 3093  df-ral 3112  df-rex 3113  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-pw 4461  df-sn 4479  df-pr 4481  df-uni 4752  df-iun 4833  df-top 21190
This theorem is referenced by: (None)
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