Minimal Element of Chain is Smallest Element

Theorem
Let $\struct {S, \preceq}$ be an ordered set.

Let $C$ be a chain in $S$.

Let $m$ be a minimal element of $C$.

Then $m$ is the smallest element of $C$.

Proof
Let $x \in C$.

Since $m$ is minimal in $C$, $x \nprec m$.

Since $C$ is a chain, $x = m$ or $m \prec x$.

Thus for each $x \in C$, $m \preceq x$.

Therefore $m$ is the smallest element of $C$.

Also see

 * Maximal Element of Chain is Greatest Element