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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 7756; an alternate proof uses indiscrete topologies (see indistop 22990) and the analogue of pwnex 7756 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7754). (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
topnex | ⊢ Top ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnex 7756 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V | |
2 | 1 | neli 3038 | . . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V |
3 | distop 22983 | . . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top) | |
4 | 3 | elv 3468 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top |
5 | eleq1 2814 | . . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top)) | |
6 | 4, 5 | mpbiri 257 | . . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
7 | 6 | exlimiv 1926 | . . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
8 | 7 | abssi 4063 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top |
9 | ssexg 5318 | . . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) | |
10 | 8, 9 | mpan 688 | . . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) |
11 | 2, 10 | mto 196 | . 2 ⊢ ¬ Top ∈ V |
12 | 11 | nelir 3039 | 1 ⊢ Top ∉ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2703 ∉ wnel 3036 Vcvv 3462 ⊆ wss 3946 𝒫 cpw 4597 Topctop 22880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-ext 2697 ax-sep 5294 ax-pow 5359 ax-pr 5423 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-un 3951 df-in 3953 df-ss 3963 df-pw 4599 df-sn 4624 df-pr 4626 df-uni 4906 df-iun 4995 df-top 22881 |
This theorem is referenced by: (None) |
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