Minimal Element of Chain is Smallest Element
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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 $C$.
$\blacksquare$
Also see
Sources
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S\text I.3$