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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 7345; an alternate proof uses indiscrete topologies (see indistop 21298) and the analogue of pwnex 7345 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7343). (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
topnex | ⊢ Top ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnex 7345 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V | |
2 | 1 | neli 3094 | . . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V |
3 | distop 21291 | . . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top) | |
4 | 3 | elv 3445 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top |
5 | eleq1 2872 | . . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top)) | |
6 | 4, 5 | mpbiri 259 | . . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
7 | 6 | exlimiv 1912 | . . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
8 | 7 | abssi 3973 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top |
9 | ssexg 5125 | . . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) | |
10 | 8, 9 | mpan 686 | . . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) |
11 | 2, 10 | mto 198 | . 2 ⊢ ¬ Top ∈ V |
12 | 11 | nelir 3095 | 1 ⊢ Top ∉ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1525 ∃wex 1765 ∈ wcel 2083 {cab 2777 ∉ wnel 3092 Vcvv 3440 ⊆ wss 3865 𝒫 cpw 4459 Topctop 21189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-nel 3093 df-ral 3112 df-rex 3113 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-pw 4461 df-sn 4479 df-pr 4481 df-uni 4752 df-iun 4833 df-top 21190 |
This theorem is referenced by: (None) |
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