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Theorem topnex 12182
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4340. (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 4340 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
21neli 2382 . . 3 ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3 vex 2663 . . . . . . . 8 𝑥 ∈ V
4 distop 12181 . . . . . . . 8 (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
53, 4ax-mp 5 . . . . . . 7 𝒫 𝑥 ∈ Top
6 eleq1 2180 . . . . . . 7 (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
75, 6mpbiri 167 . . . . . 6 (𝑦 = 𝒫 𝑥𝑦 ∈ Top)
87exlimiv 1562 . . . . 5 (∃𝑥 𝑦 = 𝒫 𝑥𝑦 ∈ Top)
98abssi 3142 . . . 4 {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
10 ssexg 4037 . . . 4 (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
119, 10mpan 420 . . 3 (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
122, 11mto 636 . 2 ¬ Top ∈ V
1312nelir 2383 1 Top ∉ V
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wnel 2380  Vcvv 2660  wss 3041  𝒫 cpw 3480  Topctop 12091
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-nel 2381  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-uni 3707  df-iun 3785  df-top 12092
This theorem is referenced by: (None)
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