To explain my point consider the set $\{ 1\} \subset \mathbb{R}$. I claim that the supremum of this set is $1$. If $x<1$, then it is not an upper bound of the set as, there is something in the set that is greater than $x$. And since we want the "smallest upper bound" no element greater than $1$ can be the sup. We are then left with $1$ as a possible canditate, and we use the supremum axiom to justify saying that the supremum is $1$.
Hence, my question is, is the usage of the supremum axiom inescapably when you want to say that something is a supremum of a set?