Proof that an infinite set can be written as the countably infinite union of pairwise disjoint infinite subsets without using the axiom of choice.

Question

I have the following theorem

Any infinite set can be written as the countably infinite union of pairwise disjoint infinite subsets

I have found a couple of proofs for this but I was wondering if there is some proof that doesn't use the axiom of choice (and it's equivalent statements). Does there exist such a proof?

Answer 1

No. An amorphous set is infinite but cannot be written even as the union of two disjoint infinite sets, and it is consistent with $\mathsf{ZF}$ without choice that amorphous sets exist.