Irrational Numbers are Uncountably Infinite

Theorem
The set $\R \setminus \Q$ of irrational numbers is uncountable.

Proof
From Real Numbers are Uncountable, $\R$ is an uncountable set.

From Rational Numbers are Countably Infinite $\Q$ is countable.

The result follows from Uncountable Set less Countable Set is Uncountable.