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| Mirrors > Home > MPE Home > Th. List > connclo | Structured version Visualization version GIF version | ||
| Description: The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.) |
| Ref | Expression |
|---|---|
| isconn.1 | ⊢ 𝑋 = ∪ 𝐽 |
| connclo.1 | ⊢ (𝜑 → 𝐽 ∈ Conn) |
| connclo.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
| connclo.3 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
| connclo.4 | ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) |
| Ref | Expression |
|---|---|
| connclo | ⊢ (𝜑 → 𝐴 = 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | connclo.3 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
| 2 | 1 | neneqd 2945 | . 2 ⊢ (𝜑 → ¬ 𝐴 = ∅) |
| 3 | connclo.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽) | |
| 4 | connclo.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) | |
| 5 | 3, 4 | elind 4200 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽))) |
| 6 | connclo.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
| 7 | isconn.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
| 8 | 7 | isconn 23421 | . . . . . . 7 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
| 9 | 8 | simprbi 496 | . . . . . 6 ⊢ (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) |
| 10 | 6, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) |
| 11 | 5, 10 | eleqtrd 2843 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋}) |
| 12 | elpri 4649 | . . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) | |
| 13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) |
| 14 | 13 | ord 865 | . 2 ⊢ (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋)) |
| 15 | 2, 14 | mpd 15 | 1 ⊢ (𝜑 → 𝐴 = 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ∅c0 4333 {cpr 4628 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 Clsdccld 23024 Conncconn 23419 |
| 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-ext 2708 |
| 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-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-conn 23420 |
| This theorem is referenced by: conndisj 23424 cnconn 23430 connsubclo 23432 t1connperf 23444 txconn 23697 connpconn 35240 cvmliftmolem2 35287 cvmlift2lem12 35319 mblfinlem1 37664 |
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