Proving order in real number set Can we prove such a statement, or is it axiomatic ? 
$$
\forall (x,y,z) \in \mathbb{R}^3 ∶(x \leq y)∧(y \leq z)⇒(x \leq z)
$$

The relation $\leq$ on the set of real numbers is the paradigmatic relation defining a partial order, because it is reflexive, antisymmetric, and transitive. (Indeed, the relation $\leq$ on $\mathbb R$ is a total order.)
The property you post asserts that $\leq$ is transitive on $\mathbb R$.
See also: Real numbers, axiomatic approach.
