I would like to know if the following proof for showing that $a = |a|$ implies $0 \leq a$ for all $a \in \mathbb R$ is correct, using the axioms of the inequality operation "$\leq$".
"Let $a \in \mathbb R$, and suppose that $a = |a|$. Assume by contradiction that $a \lt 0$. Then by definition, $|a| = -a$. Thus, it must be that $a = -a$. But this is false since $a \neq 0$. Thus, it must be that $0 \leq a$".
Thanks in advance.