Let $X \in \text{Random}(\mathbb{R})$ be a D6096: Continuous random real number.
Let $\mathbb{I}$ be the D370: Set of irrational numbers.

Using result R5362: Probability that a continuous random real number takes value in the rational numbers is zero and disjointness, we have \begin{equation} 1 = \mathbb{P}(X \in \mathbb{R}) = \mathbb{P}(X \in \mathbb{Q} \text{ or } X \in \mathbb{I}) = \mathbb{P}(X \in \mathbb{Q}) + \mathbb{P}(X \in \mathbb{I}) = \mathbb{P}(X \in \mathbb{I}) \end{equation} $\square$