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, $g = h$.