Smallest Element is Unique/Class Theory

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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 $s$ with respect to $\RR$.

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, $a = b$.

$\blacksquare$


Also see


Sources