Empty Set is Well-Ordered

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Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a relation on $S$.

Let $\O$ denote the empty set.

Let $\RR_\O$ denote the restriction of $\RR$ to $\O$.

Then $\struct {\O, \RR_\O}$ is a well-ordered set.

Proof 1

We have that $\O$ is well-ordered under $\RR$ if and only if every non-empty subset of $\O$ has a smallest element under $\RR$.

But $\O$ has no non-empty subset.

Hence this condition is satisfied vacuously.

The result follows.

$\blacksquare$

Proof 2

Let $V$ be a basic universe.

By definition of basic universe, $\O$ is an element of $V$.

By the Axiom of Transitivity, $\O$ is a class.

The result follows from Empty Class is Well-Ordered.

$\blacksquare$