Minimum of Two Elements of a Totally Ordered Set

Let $S$ be a set equipped with a total order $\le$, and let $a,b\in S$.

The minimum of $a$ and $b$, written $\min\{a,b\}$, is the element of $S$ defined as follows: if $a\le b$, then $\min\{a,b\}$ is $a$; otherwise $\min\{a,b\}$ is $b$.