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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.
StepHypRef Expression
1 pwnex 7756 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 3038 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 distop 22983 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
43elv 3468 . . . . . . 7 𝒫 𝑥 ∈ Top
5 eleq1 2814 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
64, 5mpbiri 257 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
76exlimiv 1926 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87abssi 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)
108, 9mpan 688 . . 3 (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
112, 10mto 196 . 2 ¬ Top ∈ V
1211nelir 3039 1 Top ∉ V
Colors of variables: wff setvar class
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)
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