Prove that a smallest element, if it exists, is determined uniquely.
This is follows directly for definition. An element $a \in X$ is called the smallest element of $(X, \preceq)$ if for every $x \in X$ we have $a \preceq x$. Suppose was some other element $b \in X$ for which for every $x \in X$ we would have $b \preceq x$, but then $a \preceq b$ could not hold, unless $a=b$.
Is this really simple proof correct?