No, it is not. In particular, notice that none of the other axioms will ever allow you to construct a set with cardinality greater than $\mathbb N$. For most of the axioms, this is fairly obvious. The only particularly dangerous one is the axiom of union; however, this axiom cannot necessarily create larger sets than it is given - in particular, if we only have countable sets, then we can only create the countable union of countable sets through axiom of union (since the set over which we take the union must be countable, as must each element thereof), which, assuming AC, is countable. Therefore, the axiom of powerset, which implies that there are sets larger than $\mathbb N$ must not follow from the other axioms.