I want to express "$x$ is rational and not equal to $0$"

$x\neq 0 \in \mathbb{Q}$ seems awkward to me since $x\neq 0$ is a logical statement, not a rational number. Options I can think of include:
$$x\in \mathbb{Q} \setminus 0$$
$$x\in \mathbb{Q} \land x \neq 0$$
$$x \in \mathbb{Q} | x\neq 0$$

Is there a standard convention? I'm trying to develop a sensible, standard notation to use amongst my classes.