Between two Rational Numbers exists Irrational Number/Lemma 2

Lemma for Between Every Two Rationals Exists an Irrational
Let $\alpha \in \Q$ and $\beta \in \R \setminus \Q$.

Then:
 * $\alpha + \beta \in \R \setminus \Q$

Proof
Aiming for a contradiction, suppose that $\alpha + \beta \in \Q$.

By the definition of rational numbers:
 * $\exists p, q \in \Z: \alpha + \beta = \dfrac p q$

Thus:
 * $\beta = \dfrac p q - \alpha$

By Rational Subtraction is Closed, we have $\beta \in \Q$, which contradicts the statement that $\beta \in \R \setminus \Q$.

Therefore:
 * $\alpha + \beta \in \R \setminus \Q$