Greatest Element is Unique/Class Theory

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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.

That is, if $g$ and $h$ are both smallest elements of $A$, then $g = h$.


Let $g$ and $h$ both be smallest elements of $A$.

Then by definition:

$\forall y \in A: y \mathrel \RR g$
$\forall y \in A: y \mathrel \RR h$

Thus it follows that:

$g \preceq h$
$h \preceq g$

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

Hence by definition of antisymmetric, $g = h$.


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