Positivity and Symmetry from defination of Metric

Question

I can across definition of metric as

X is set

$d:X\times X\to \mathbb R$

satisfying 1) $d(x,y)=0 $ iff $x=y$

2) $d(x,y)\leq d(x,z)+d(y,z) , \forall x,y,z\in X$

I wanted to prove positivity and symmetry form above assumption

I can prove positivity if I have assumption 1 ,2 and symmetric.

But I could not proceed to prove both with just 2 assumption

Any Hint will be appreciated

Thanking You

Answer 1

$0 = d(x,x) \leq d(x,y) + d(x,y) = 2d(x,y)$, so $d(x,y)\geq 0$.

$d(x,y) \leq d(x,x) + d(y,x) = d(y,x)$, $d(y,x) \leq d(y,y) + d(x,y) = d(x,y)$.

So $d(x,y) = d(y,x)$.