It is not the case that AC is provable in ZF.
Meanwhile, there are two senses of "true" we would consider in this question. One is that of a formal notion of truth, i.e. "true in a given model of ZF". The other is that of an informal notion of what is basically true about sets. As to the formal notion, AC is true in some models of ZF and AC is false in other models of ZF. As to the informal notion, many people DO consider AC to be quite apparently true. Yet, (probably) a small percentage of mathematicians consider AC to be false (note that in set theory without the logical principle of excluded middle, AC implies the principle of excluded middle).