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

Definition df-con 21025
 Description: Topologies are connected when only ∅ and ∪ 𝑗 are both open and closed. (Contributed by FL, 17-Nov-2008.)
Assertion
Ref Expression
df-con Con = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}

Detailed syntax breakdown of Definition df-con
StepHypRef Expression
1 ccon 21024 . 2 class Con
2 vj . . . . . 6 setvar 𝑗
32cv 1474 . . . . 5 class 𝑗
4 ccld 20630 . . . . . 6 class Clsd
53, 4cfv 5804 . . . . 5 class (Clsd‘𝑗)
63, 5cin 3539 . . . 4 class (𝑗 ∩ (Clsd‘𝑗))
7 c0 3874 . . . . 5 class
83cuni 4372 . . . . 5 class 𝑗
97, 8cpr 4127 . . . 4 class {∅, 𝑗}
106, 9wceq 1475 . . 3 wff (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}
11 ctop 20517 . . 3 class Top
1210, 2, 11crab 2900 . 2 class {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
131, 12wceq 1475 1 wff Con = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
 Colors of variables: wff setvar class This definition is referenced by:  iscon  21026
 Copyright terms: Public domain W3C validator