![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2735 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | isconn 23437 | . 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 1537 ∈ wcel 2106 ∩ cin 3962 ∅c0 4339 {cpr 4633 ∪ cuni 4912 ‘cfv 6563 Topctop 22915 Clsdccld 23040 Conncconn 23435 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-conn 23436 |
This theorem is referenced by: conncompss 23457 txconn 23713 qtopconn 23733 ufildr 23955 connpconn 35220 cvmliftmolem1 35266 cvmliftmolem2 35267 ordtopconn 36422 |
Copyright terms: Public domain | W3C validator |