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| Mirrors > Home > MPE Home > Th. List > conntop | Structured version Visualization version GIF version | ||
| Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.) | 
| Ref | Expression | 
|---|---|
| conntop | ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | 1 | isconn 23422 | . 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, ∪ 𝐽})) | 
| 3 | 2 | simplbi 497 | 1 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∩ cin 3949 ∅c0 4332 {cpr 4627 ∪ cuni 4906 ‘cfv 6560 Topctop 22900 Clsdccld 23025 Conncconn 23420 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-conn 23421 | 
| This theorem is referenced by: conncompss 23442 txconn 23698 qtopconn 23718 ufildr 23940 connpconn 35241 cvmliftmolem1 35287 cvmliftmolem2 35288 ordtopconn 36441 | 
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