Separable Discrete Space is Countable/Proof 2

Theorem

Let $T = \struct {S, \tau}$ be a discrete topological space.

Let $T$ be separable.

Then $S$ is countable.

Proof

Let $T$ be separable.

Aiming for a contradiction, suppose $S$ is uncountable.

Then by Uncountable Discrete Space is not Separable, $T$ is not separable.

Hence the result by Proof by Contradiction.

$\blacksquare$