Minimal Element of Chain is Smallest Element

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Theorem

Let $\left({S, \preceq}\right)$ 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 $C$, $x \not\prec m$.

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 $C$.

$\blacksquare$


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