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Theorem bj-topnex 32247
 Description: The class of all topologies is a proper class. (Contributed by BJ, 2-May-2021.)
Assertion
Ref Expression
bj-topnex Top ∉ V

Proof of Theorem bj-topnex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-pwnex 32246 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 2885 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 vex 3176 . . . . . . . 8 𝑥 ∈ V
4 distop 20610 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
53, 4ax-mp 5 . . . . . . 7 𝒫 𝑥 ∈ Top
6 eleq1 2676 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
75, 6mpbiri 247 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87exlimiv 1845 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
98abssi 3640 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10 ssexg 4732 . . . 4 (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
119, 10mpan 702 . . 3 (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
122, 11mto 187 . 2 ¬ Top ∈ V
1312nelir 2886 1 Top ∉ V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ∉ wnel 2781  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  Topctop 20517 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-top 20521 This theorem is referenced by: (None)
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