Smallest Element is Unique

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Theorem

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

If $S$ has a smallest element, then it can have only one.


That is, if $a$ and $b$ are both smallest elements of $S$, then $a = b$.


Class Theoretical Formulation

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be an ordering.

Let $A$ be a subclass of the field of $\RR$.


Suppose $A$ has a smallest element $s$ with respect to $\RR$.

Then $s$ is unique.


Proof

Let $a$ and $b$ both be smallest elements of $S$.

Then by definition:

$\forall y \in S: a \preceq y$
$\forall y \in S: b \preceq y$

Thus it follows that:

$a \preceq b$
$b \preceq a$

But as $\preceq$ is an ordering, it is antisymmetric.

Hence by definition of antisymmetric, $a = b$.

$\blacksquare$


Also see


Sources

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