Non-Trivial Discrete Space is not Ultraconnected

Theorem
Let $T = \left({S, \tau}\right)$ be a non-trivial discrete topological space.

$T$ is not ultraconnected.

Proof
$T$ is ultraconnected.

From Ultraconnected Space is Connected, we have that $T$ is connected.

But this directly contradicts Non-Trivial Discrete Space is not Connected.

The result follows from Proof by Contradiction.