Axiom:Axiom of Choice for Finite Sets/Proof from Ordering Principle

Theorem
Suppose that the Ordering Principle holds.

Let $\mathcal S$ be a non-empty family of finite, non-empty sets.

Then there exists a choice function for $\mathcal S$.

Proof from the Ordering Principle
By the Axiom of Union, $\mathcal S$ has a union.

Let $U = \bigcup \mathcal S$.

By the Ordering Principle, there is a total ordering $\preceq$ on $U$.

For each $S \in \mathcal S$, $S$ is a chain in $U$.

By Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements, each $S \in S$ has a minimum.

Let $f:\mathcal S \to U$ be defined by:
 * $f(S) = \min S$

Then $f$ is a choice function for $\mathcal S$.