Unions and the axiom of choice.

The answer is that we don't know how to prove that this is equivalent to the axiom of choice.

Higasikawa, Masasi "Partition principles and infinite sums of cardinal numbers." Notre Dame J. Formal Logic 36 (1995), no. 3, 425–434.

In the above paper the author shows that this principle (which he names $\sf FB$) implies the partition principle (if there is a surjection from $A$ onto $B$, then there is an injection from $B$ into $A$). The question whether or not the partition principle implies the axiom of choice is open.

You can also find this principle in Gregory Moore's book about the axiom of choice. This is principle 1.4.12, and it is indicated to be open whether or not it is equivalent to the axiom of choice. However, it is also indicated that in conjunction with the following principle, we can prove the axiom of choice:

"If every member of an infinite set $A$ has the same cardinality of $A$ then $\bigcup A$ has the same cardinality as $A$."

So all in all, this question is another open question from the list of "very easy to formulate, very difficult to prove equivalent to the axiom of choice".