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Theorem topnex 21014
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7198; an alternate proof uses indiscrete topologies (see indistop 21020) and the analogue of pwnex 7198 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7195). (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 7198 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 3083 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 vex 3394 . . . . . . . 8 𝑥 ∈ V
4 distop 21013 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
53, 4ax-mp 5 . . . . . . 7 𝒫 𝑥 ∈ Top
6 eleq1 2873 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
75, 6mpbiri 249 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87exlimiv 2021 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
98abssi 3874 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10 ssexg 4999 . . . 4 (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
119, 10mpan 673 . . 3 (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
122, 11mto 188 . 2 ¬ Top ∈ V
1312nelir 3084 1 Top ∉ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wnel 3081  Vcvv 3391  wss 3769  𝒫 cpw 4351  Topctop 20911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-nel 3082  df-ral 3101  df-rex 3102  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-pw 4353  df-sn 4371  df-pr 4373  df-uni 4631  df-iun 4714  df-top 20912
This theorem is referenced by: (None)
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