![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > topnex | GIF version |
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4449. (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
topnex | ⊢ Top ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnex 4449 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V | |
2 | 1 | neli 2444 | . . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V |
3 | vex 2740 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | distop 13478 | . . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top |
6 | eleq1 2240 | . . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top)) | |
7 | 5, 6 | mpbiri 168 | . . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
8 | 7 | exlimiv 1598 | . . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
9 | 8 | abssi 3230 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top |
10 | ssexg 4142 | . . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) | |
11 | 9, 10 | mpan 424 | . . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) |
12 | 2, 11 | mto 662 | . 2 ⊢ ¬ Top ∈ V |
13 | 12 | nelir 2445 | 1 ⊢ Top ∉ V |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 ∉ wnel 2442 Vcvv 2737 ⊆ wss 3129 𝒫 cpw 3575 Topctop 13388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-nel 2443 df-ral 2460 df-rex 2461 df-v 2739 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-uni 3810 df-iun 3888 df-top 13389 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |