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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering