A Minimum in a Total Ordering is Unique

Last updated Nov 1, 2022

Suppose $(T, \leq)$ is a Total Ordering. Let $A \subset T$. Then if $x \in A$ is a minimal element of $A$, $x$ is the only Minimum of $A$.

Let $x, x’ \in A$ be minimal elements. Since $T$ is a total ordering, we can compare $x$ and $x’$. Since $x$ is a Minimum, we know $x \leq x’$. However $x’$ is also a Minimum, so $x’ \leq x$. By definition of an Order Relation, we have that $x’ = x$ and the Minimum is unique. $\blacksquare$