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Q:

Are all definition axioms for order structures also true for ultrametric ones?

Let $(X,\tau)$ be a non-empty set equipped with an ultra-metric $\tau$, i.e. a metric for which $d_{\tau}(x,y)=0$ iff $x=y$.
Two ultra-metrics are ultrametric, if the ultrametric inequality $d_{\tau}(x,y)\leq\max\{d_{\tau}(x,z),d_{\tau}(z,y)\}$ always hold.

Are all axioms for order structures also true for ultrametric ones? (That is:

$\tau(x,y)=\tau(x,z)+\tau(z,y)$
$\tau(x,y)+\tau(y,x)=d_{\tau}(x,y)$
$\tau(x,x)=0$
$\tau(x,y)=0$ iff $x=y$
$\tau(x,y)=\tau(y,x)$
If $\tau(x,y)=0$ and $x eq y$, then $x\leq y$)

Note that (5) is not very natural, if the order is ultrametric, as it requires to prove that it

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