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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 3021 | . 2 ⊢ (𝜑 → ¬ 𝐴 = ∅) |
3 | connclo.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽) | |
4 | connclo.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) | |
5 | 3, 4 | elind 4170 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽))) |
6 | connclo.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
7 | isconn.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
8 | 7 | isconn 22020 | . . . . . . 7 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
9 | 8 | simprbi 499 | . . . . . 6 ⊢ (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) |
10 | 6, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) |
11 | 5, 10 | eleqtrd 2915 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋}) |
12 | elpri 4588 | . . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) |
14 | 13 | ord 860 | . 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 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∩ cin 3934 ∅c0 4290 {cpr 4568 ∪ cuni 4837 ‘cfv 6354 Topctop 21500 Clsdccld 21623 Conncconn 22018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-conn 22019 |
This theorem is referenced by: conndisj 22023 cnconn 22029 connsubclo 22031 t1connperf 22043 txconn 22296 connpconn 32482 cvmliftmolem2 32529 cvmlift2lem12 32561 mblfinlem1 34928 |
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