Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-conn Structured version   Visualization version   GIF version

Definition df-conn 22021
 Description: Topologies are connected when only ∅ and ∪ 𝑗 are both open and closed. (Contributed by FL, 17-Nov-2008.)
Assertion
Ref Expression
df-conn Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}

Detailed syntax breakdown of Definition df-conn
StepHypRef Expression
1 cconn 22020 . 2 class Conn
2 vj . . . . . 6 setvar 𝑗
32cv 1537 . . . . 5 class 𝑗
4 ccld 21625 . . . . . 6 class Clsd
53, 4cfv 6328 . . . . 5 class (Clsd‘𝑗)
63, 5cin 3883 . . . 4 class (𝑗 ∩ (Clsd‘𝑗))
7 c0 4246 . . . . 5 class
83cuni 4803 . . . . 5 class 𝑗
97, 8cpr 4530 . . . 4 class {∅, 𝑗}
106, 9wceq 1538 . . 3 wff (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}
11 ctop 21502 . . 3 class Top
1210, 2, 11crab 3113 . 2 class {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
131, 12wceq 1538 1 wff Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
 Colors of variables: wff setvar class This definition is referenced by:  isconn  22022
 Copyright terms: Public domain W3C validator