A consequence of booleanity of the category of Set?

Question

In Sheaves in Geometry and Logic one can find the following theorem:

Let $\mathcal{E}$ be an elementary topos which is generated by subobjects of $1$, and moreover has the property that for each object $E$, Sub$(E)$ is a complete Boolan algebra. Then $\mathcal{E}$ satisfies the axiom of choice (every epi splits).

Now, the category of Sets satisfies these two hypothesis but in general it does not verify the axiom of choice. What is wrong?!

Answer 1

You have to be careful about the background theory: As you can see in the proof of the theorem, Zorn's Lemma is used, so the underlying set theory must include the axiom of choice here. In particular, the statement will not hold in the first-order theory of elementary toposes.