Uncountable Set less Countable Set is Uncountable

Theorem
Let $S$ be an uncountable set.

Let $T \subseteq S$ be a countable subset of $S$.

Then:
 * $S \setminus T$ is uncountable

where $\setminus$ denotes set difference.

Proof
Suppose $S \setminus T$ were countable.

By definition of relative complement:
 * $S \setminus T = \complement_S \left({T}\right)$

Thus from Union with Relative Complement:
 * $\left({S \setminus T}\right) \cup T = S$

But from Finite Union of Countable Sets is Countable it follows that $S$ is countable.

From this contradiction it follows that $S \setminus T$ is uncountable.