Countable Union of Finite Sets is Countable

Theorem
The following statements are equivalent in $\mathrm{ZF}^-$:


 * The Axiom of Countable Choice for Finite Sets holds.


 * The union of any countable set of finite sets is countable.

Axiom of Countable Choice for Finite Sets implies Countable Union Condition for Finite Sets
Suppose that the Axiom of Countable Choice for Finite Sets holds.

Let $\mathcal F$ be a countable set of finite sets.

Then $\mathcal F$ is either finite or countably infinite.

If $\mathcal F$ is finite, then $\bigcup \mathcal F$ is finite by Finite Union of Finite Sets is Finite, and thus countable.

Suppose instead that $\mathcal F$ is countably infinite.

Then there is a bijection $f: \N \to \mathcal F$.