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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 2942 | . 2 ⊢ (𝜑 → ¬ 𝐴 = ∅) |
3 | connclo.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽) | |
4 | connclo.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) | |
5 | 3, 4 | elind 4209 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽))) |
6 | connclo.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
7 | isconn.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
8 | 7 | isconn 23436 | . . . . . . 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 2840 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋}) |
12 | elpri 4653 | . . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) |
14 | 13 | ord 864 | . 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 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∩ cin 3961 ∅c0 4338 {cpr 4632 ∪ cuni 4911 ‘cfv 6562 Topctop 22914 Clsdccld 23039 Conncconn 23434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-conn 23435 |
This theorem is referenced by: conndisj 23439 cnconn 23445 connsubclo 23447 t1connperf 23459 txconn 23712 connpconn 35219 cvmliftmolem2 35266 cvmlift2lem12 35298 mblfinlem1 37643 |
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