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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 4340. (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
topnex | ⊢ Top ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnex 4340 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V | |
2 | 1 | neli 2382 | . . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V |
3 | vex 2663 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | distop 12181 | . . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top |
6 | eleq1 2180 | . . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top)) | |
7 | 5, 6 | mpbiri 167 | . . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
8 | 7 | exlimiv 1562 | . . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
9 | 8 | abssi 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) | |
11 | 9, 10 | mpan 420 | . . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) |
12 | 2, 11 | mto 636 | . 2 ⊢ ¬ Top ∈ V |
13 | 12 | nelir 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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