Smallest 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 smallest element $s$ $\RR$.

Then $s$ is unique.

That is, if $s$ and $t$ are both smallest elements of $A$, then $s = t$.

Proof
Let $s$ and $t$ both be smallest elements of $A$.

Then by definition:
 * $\forall y \in A: s \mathrel \RR y$
 * $\forall y \in A: t \mathrel \RR y$

Thus it follows that:
 * $s \preceq t$
 * $t \preceq s$

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

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

Also see

 * Greatest Element of Class is Unique