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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 2946 | . 2 ⊢ (𝜑 → ¬ 𝐴 = ∅) |
3 | connclo.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽) | |
4 | connclo.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) | |
5 | 3, 4 | elind 4195 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽))) |
6 | connclo.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
7 | isconn.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
8 | 7 | isconn 22917 | . . . . . . 7 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
9 | 8 | simprbi 498 | . . . . . 6 ⊢ (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) |
10 | 6, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) |
11 | 5, 10 | eleqtrd 2836 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋}) |
12 | elpri 4651 | . . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) |
14 | 13 | ord 863 | . 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 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∩ cin 3948 ∅c0 4323 {cpr 4631 ∪ cuni 4909 ‘cfv 6544 Topctop 22395 Clsdccld 22520 Conncconn 22915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-conn 22916 |
This theorem is referenced by: conndisj 22920 cnconn 22926 connsubclo 22928 t1connperf 22940 txconn 23193 connpconn 34257 cvmliftmolem2 34304 cvmlift2lem12 34336 mblfinlem1 36573 |
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