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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 398 | . . 3 ⊢ (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅})) | |
2 | 0opn 20882 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
3 | n0i 4051 | . . . . . 6 ⊢ (∅ ∈ 𝐽 → ¬ 𝐽 = ∅) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ Top → ¬ 𝐽 = ∅) |
5 | 4 | pm2.21d 118 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 = ∅ → 𝐽 = {∅})) |
6 | idd 24 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} → 𝐽 = {∅})) | |
7 | 5, 6 | jaod 394 | . . 3 ⊢ (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅})) |
8 | 1, 7 | impbid2 216 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))) |
9 | uni0b 4603 | . . 3 ⊢ (∪ 𝐽 = ∅ ↔ 𝐽 ⊆ {∅}) | |
10 | sssn 4491 | . . 3 ⊢ (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})) | |
11 | 9, 10 | bitr2i 265 | . 2 ⊢ ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ ∪ 𝐽 = ∅) |
12 | 8, 11 | syl6rbb 277 | 1 ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 = wceq 1620 ∈ wcel 2127 ⊆ wss 3703 ∅c0 4046 {csn 4309 ∪ cuni 4576 Topctop 20871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ral 3043 df-rex 3044 df-v 3330 df-dif 3706 df-in 3710 df-ss 3717 df-nul 4047 df-pw 4292 df-sn 4310 df-uni 4577 df-top 20872 |
This theorem is referenced by: locfinref 30188 |
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