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Theorem topnex 12880
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4434. (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 4434 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 2437 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 vex 2733 . . . . . . . 8 𝑥 ∈ V
4 distop 12879 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
53, 4ax-mp 5 . . . . . . 7 𝒫 𝑥 ∈ Top
6 eleq1 2233 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
75, 6mpbiri 167 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87exlimiv 1591 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
98abssi 3222 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10 ssexg 4128 . . . 4 (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
119, 10mpan 422 . . 3 (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
122, 11mto 657 . 2 ¬ Top ∈ V
1312nelir 2438 1 Top ∉ V
Colors of variables: wff set class
Syntax hints:   = wceq 1348  wex 1485  wcel 2141  {cab 2156  wnel 2435  Vcvv 2730  wss 3121  𝒫 cpw 3566  Topctop 12789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-nel 2436  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-uni 3797  df-iun 3875  df-top 12790
This theorem is referenced by: (None)
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