Non-Trivial Discrete Space is not Path-Connected

Corollary to Non-Trivial Discrete Space is not Connected
Let $T = \struct {S, \tau}$ be a non-trivial discrete topological space.

$T$ is not path-connected.

Proof
$T$ is path-connected.

From Path-Connected 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.