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Mirrors > Home > MPE Home > Th. List > topnex | Structured version Visualization version GIF version |
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7778; an alternate proof uses indiscrete topologies (see indistop 23025) and the analogue of pwnex 7778 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7776). (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
topnex | ⊢ Top ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnex 7778 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V | |
2 | 1 | neli 3046 | . . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V |
3 | distop 23018 | . . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top) | |
4 | 3 | elv 3483 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top |
5 | eleq1 2827 | . . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top)) | |
6 | 4, 5 | mpbiri 258 | . . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
7 | 6 | exlimiv 1928 | . . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
8 | 7 | abssi 4080 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top |
9 | ssexg 5329 | . . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) | |
10 | 8, 9 | mpan 690 | . . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) |
11 | 2, 10 | mto 197 | . 2 ⊢ ¬ Top ∈ V |
12 | 11 | nelir 3047 | 1 ⊢ Top ∉ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ∉ wnel 3044 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 Topctop 22915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-un 3968 df-in 3970 df-ss 3980 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-iun 4998 df-top 22916 |
This theorem is referenced by: (None) |
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