A D4: Binary relation ${\preceq} \subseteq X \times X$ on $X$ is a partial ordering relation on $X$ if and only if

(1)	$\forall \, x \in X : (x, x) \in {\preceq}$ (D287: Reflexive binary relation)
(2)	$\forall \, x, y, z \in X \, ((x, y), (y, z) \in {\preceq} \quad \implies \quad (x, z) \in {\preceq})$ (D288: Transitive binary relation)
(3)	$\forall \, x, y \in X \, ((x, y), (y, x) \in {\preceq} \quad \implies \quad x = y)$ (D289: Antisymmetric binary relation)

A D4: Binary relation ${\preceq} \subseteq X \times X$ is a partial ordering relation on $X$ if and only if

(1)	$\forall \, x \in X : x \preceq x$ (D287: Reflexive binary relation)
(2)	$\forall \, x, y, z \in X \, (x \preceq y \preceq z \quad \implies \quad x \preceq z)$ (D288: Transitive binary relation)
(3)	$\forall \, x, y \in X \, (x \preceq y \text{ and } y \preceq x \quad \implies \quad x = y)$ (D289: Antisymmetric binary relation)