Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
Let $\mathsf{UB} = \mathsf{UB}_P(E)$ be the D552: Set of upper bounds of $E \subseteq X$ with respect to $P$.
A D2218: Set element $x_0 \in X$ is a supremum of $E$ with respect to $P$ if and only if

(1)	$x_0 \in \mathsf{UB}$
(2)	$\forall \, x \in \mathsf{UB} : (x_0, x) \in {\preceq}$

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
Let $\mathsf{UB} = \mathsf{UB}_P(E)$ be the D552: Set of upper bounds of $E \subseteq X$ with respect to $P$.
A D2218: Set element $x_0 \in X$ is a supremum of $E$ with respect to $P$ if and only if

(1)	$x_0 \in \mathsf{UB}$
(2)	$\forall \, x \in \mathsf{UB} : x_0 \preceq x$