Using the order axioms of $\mathbb{R}$ to prove the semi-definite positivity property for the absolute value?

Question

How can I use the order axioms of $\mathbb{R}$ to prove the semi-definite positivity property for the absolute value:

For all $x \in \mathbb{R}, |x|\geq0$ and $|x|=0$ if and only if $x=0$?

Answer 1

If 0 <= x, then -x <= 0. Thus |x| = max(x,-x) = x >= 0.
If x <= 0, then 0 <= -x. Thus |x| = max(x,-x) = -x >= 0.

If |x| = 0, then x = 0 or -x = 0. Thus x = 0.
If x = 0, then |x| = 0.