Metric Space Axioms
Solution 1:
There is obviously no need ;
But since "Metric Space" deals with distance function and the distance between any two objects is always non-negative so the axiom is added to mark its significance.
Solution 2:
In fact, it is enough to require only that $$d(x,y)=0 \textrm{ if and only if } x=y$$ and $$d(y,z)\leq d(x,y) +d(x,z)$$
All other properties of the metric follow from these, including nonnegativity.