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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 3019 | . 2 ⊢ (𝜑 → ¬ 𝐴 = ∅) |
3 | connclo.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽) | |
4 | connclo.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) | |
5 | 3, 4 | elind 4169 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽))) |
6 | connclo.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn) | |
7 | isconn.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
8 | 7 | isconn 22013 | . . . . . . 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 2913 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋}) |
12 | elpri 4581 | . . . 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 1531 ∈ wcel 2108 ≠ wne 3014 ∩ cin 3933 ∅c0 4289 {cpr 4561 ∪ cuni 4830 ‘cfv 6348 Topctop 21493 Clsdccld 21616 Conncconn 22011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-conn 22012 |
This theorem is referenced by: conndisj 22016 cnconn 22022 connsubclo 22024 t1connperf 22036 txconn 22289 connpconn 32475 cvmliftmolem2 32522 cvmlift2lem12 32554 mblfinlem1 34921 |
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