Smallest Element is Minimal
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Theorem
Let $\struct {S, \preceq}$ be an ordered set which has a smallest element.
Let $m$ be the smallest element of $\struct {S, \preceq}$.
Then $m$ is a minimal element.
Proof
By definition of smallest element:
- $\forall y \in S: m \preceq y$
Suppose $y \preceq m$.
As $\preceq$ is an ordering, $\preceq$ is by definition antisymmetric.
Thus it follows by definition of antisymmetry that $m = y$.
That is:
- $y \preceq m \implies m = y$
which is precisely the definition of a minimal element.
$\blacksquare$
Also see
Sources
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations