![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0top | Structured version Visualization version GIF version |
Description: The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.) |
Ref | Expression |
---|---|
0top | ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 865 | . . 3 ⊢ (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅})) | |
2 | 0opn 21509 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
3 | n0i 4249 | . . . . . 6 ⊢ (∅ ∈ 𝐽 → ¬ 𝐽 = ∅) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ Top → ¬ 𝐽 = ∅) |
5 | 4 | pm2.21d 121 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 = ∅ → 𝐽 = {∅})) |
6 | idd 24 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} → 𝐽 = {∅})) | |
7 | 5, 6 | jaod 856 | . . 3 ⊢ (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅})) |
8 | 1, 7 | impbid2 229 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))) |
9 | uni0b 4826 | . . 3 ⊢ (∪ 𝐽 = ∅ ↔ 𝐽 ⊆ {∅}) | |
10 | sssn 4719 | . . 3 ⊢ (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})) | |
11 | 9, 10 | bitr2i 279 | . 2 ⊢ ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ ∪ 𝐽 = ∅) |
12 | 8, 11 | syl6rbb 291 | 1 ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 {csn 4525 ∪ cuni 4800 Topctop 21498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-uni 4801 df-top 21499 |
This theorem is referenced by: locfinref 31194 |
Copyright terms: Public domain | W3C validator |