A partial order is an antisymmetric preorder, i.e. a binary relation $\leq$ on $X$ such that:
Reflexivity. For all $x\in X$,
\[x \leq x.\]Transitivity. For all $x$, $y$, $z\in X$,
\[x\leq y \,\land\, y\leq z \enspace\rimp\enspace x\leq z.\]Antisymmetry. For all $x$, $y\in X$,
\[x\leq y \,\land\, y\leq x \enspace\rimp\enspace x=y.\]
PO#DEF-S. Strict Partial Order.
A binary relation $<$ on $X$ is a strict partial order if:
Irreflexivity. For all $x\in X$,
\[x \nless x.\]Transitivity. For all $x$, $y$, $z\in X$,
\[x<y \,\land\, y<z \enspace\rimp\enspace x<z.\]Asymmetry. For all $x$, $y\in X$,
\[x<y \enspace\rimp\enspace y\nless x.\]
A binary relation is a strict partial order as soon as it is irreflexive and transitive.
(This is why we do not talk about “strict preorder”.)
Proof.Irreflexivity and transitivity imply asymmetry: Let $<$ be an irreflexive and transitive relation on $X$. If $x<y$ and $y<x$, then $x<x$ by transitivity, contrary to irreflexivity.
$\leq$ is a partial order if and only if $<$ is a strict partial order.