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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 4194 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽))) |
6 | connclo.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
7 | isconn.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
8 | 7 | isconn 23137 | . . . . . . 7 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
9 | 8 | simprbi 497 | . . . . . 6 ⊢ (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) |
10 | 6, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) |
11 | 5, 10 | eleqtrd 2835 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋}) |
12 | elpri 4650 | . . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋)) |
14 | 13 | ord 862 | . 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 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∩ cin 3947 ∅c0 4322 {cpr 4630 ∪ cuni 4908 ‘cfv 6543 Topctop 22615 Clsdccld 22740 Conncconn 23135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-conn 23136 |
This theorem is referenced by: conndisj 23140 cnconn 23146 connsubclo 23148 t1connperf 23160 txconn 23413 connpconn 34512 cvmliftmolem2 34559 cvmlift2lem12 34591 mblfinlem1 36828 |
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