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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 22730 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
3 | n0i 4326 | . . . . . 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 225 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))) |
9 | uni0b 4928 | . . 3 ⊢ (∪ 𝐽 = ∅ ↔ 𝐽 ⊆ {∅}) | |
10 | sssn 4822 | . . 3 ⊢ (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})) | |
11 | 9, 10 | bitr2i 276 | . 2 ⊢ ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ ∪ 𝐽 = ∅) |
12 | 8, 11 | bitr2di 288 | 1 ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 ∅c0 4315 {csn 4621 ∪ cuni 4900 Topctop 22719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-ext 2695 ax-sep 5290 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 df-nul 4316 df-pw 4597 df-sn 4622 df-uni 4901 df-top 22720 |
This theorem is referenced by: locfinref 33313 |
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