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

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Suppose that the Ordering Principle holds.

Let $\SS$ be a non-empty set of finite non-empty sets.

Then there exists a choice function for $\SS$.

Proof from the Ordering Principle

By the Axiom of Union, $\SS$ has a union.

Let $U = \bigcup \SS$.

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

For each $S \in \SS$, $S$ 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 $f: \SS \to U$ be defined by:

$\map f S = \min S$

Then $f$ is a choice function for $\SS$.