Let $(X,\preceq)$ be a partially ordered set. Fix $S\subseteq X$. Say $S$ is sup-closed if whenever $A$ is a nonempty subset of $S$ whose supremum exists, then $\sup A$ is an element of $S$.
Apparently one can describe the sup-closure of $S$ as the following set: $\{x\in X:\text{$x=\sup A$ for some nonempty subset $A$ of S}\}$. I managed to find a proof, but I helped myself to the Axiom of Choice in order to prove that the described set is indeed sup-closed. I am wondering if there is an argument which does not use AC? That is to say, is there a description of the sup-closure within a poset without the Axiom of Choice?
EDIT: Of course one can describe the sup-closure as the intersection of all sup-closed sets containing $S$. I guess the main question is if the particular description written above can be verified without AC.