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Theorem topnex 22499
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7746; an alternate proof uses indiscrete topologies (see indistop 22505) and the analogue of pwnex 7746 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7744). (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 7746 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 3049 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 distop 22498 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
43elv 3481 . . . . . . 7 𝒫 𝑥 ∈ Top
5 eleq1 2822 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
64, 5mpbiri 258 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
76exlimiv 1934 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87abssi 4068 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
9 ssexg 5324 . . . 4 (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
108, 9mpan 689 . . 3 (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
112, 10mto 196 . 2 ¬ Top ∈ V
1211nelir 3050 1 Top ∉ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wnel 3047  Vcvv 3475  wss 3949  𝒫 cpw 4603  Topctop 22395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910  df-iun 5000  df-top 22396
This theorem is referenced by: (None)
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