Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  topnex Structured version   Visualization version   GIF version

Theorem topnex 21022
 Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7134; an alternate proof uses indiscrete topologies (see indistop 21028) and the analogue of pwnex 7134 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7131). (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 7134 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 3037 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 vex 3343 . . . . . . . 8 𝑥 ∈ V
4 distop 21021 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
53, 4ax-mp 5 . . . . . . 7 𝒫 𝑥 ∈ Top
6 eleq1 2827 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
75, 6mpbiri 248 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87exlimiv 2007 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
98abssi 3818 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10 ssexg 4956 . . . 4 (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
119, 10mpan 708 . . 3 (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
122, 11mto 188 . 2 ¬ Top ∈ V
1312nelir 3038 1 Top ∉ V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632  ∃wex 1853   ∈ wcel 2139  {cab 2746   ∉ wnel 3035  Vcvv 3340   ⊆ wss 3715  𝒫 cpw 4302  Topctop 20920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-nel 3036  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589  df-iun 4674  df-top 20921 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator