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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 869 | . . 3 ⊢ (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅})) | |
| 2 | 0opn 22910 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
| 3 | n0i 4340 | . . . . . 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 860 | . . 3 ⊢ (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅})) |
| 8 | 1, 7 | impbid2 226 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))) |
| 9 | uni0b 4933 | . . 3 ⊢ (∪ 𝐽 = ∅ ↔ 𝐽 ⊆ {∅}) | |
| 10 | sssn 4826 | . . 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 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∅c0 4333 {csn 4626 ∪ cuni 4907 Topctop 22899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-uni 4908 df-top 22900 |
| This theorem is referenced by: locfinref 33840 |
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