![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7746; an alternate proof uses indiscrete topologies (see indistop 22505) and the analogue of pwnex 7746 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7744). (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
topnex | ⊢ Top ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnex 7746 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V | |
2 | 1 | neli 3049 | . . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V |
3 | distop 22498 | . . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top) | |
4 | 3 | elv 3481 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top |
5 | eleq1 2822 | . . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top)) | |
6 | 4, 5 | mpbiri 258 | . . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
7 | 6 | exlimiv 1934 | . . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
8 | 7 | abssi 4068 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top |
9 | ssexg 5324 | . . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) | |
10 | 8, 9 | mpan 689 | . . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) |
11 | 2, 10 | mto 196 | . 2 ⊢ ¬ Top ∈ V |
12 | 11 | nelir 3050 | 1 ⊢ Top ∉ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ∉ wnel 3047 Vcvv 3475 ⊆ wss 3949 𝒫 cpw 4603 Topctop 22395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2704 ax-sep 5300 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-un 3954 df-in 3956 df-ss 3966 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4910 df-iun 5000 df-top 22396 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |