Greatest Element is Unique/Class Theory

Theorem
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 $g$ $\RR$.

Then $g$ is unique.

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

Proof
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, $g = h$.

Also see

 * Smallest Element of Class is Unique