Greatest Element is Unique

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Theorem

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

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


That is, if $a$ and $b$ are both greatest 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 greatest element $g$ with respect to $\RR$.

Then $g$ is unique.


Proof

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

Then by definition:

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

Thus it follows that:

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

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

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

$\blacksquare$


Also see


Sources

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