Almost everything most mathematicians do can be done both in ZFC and your favourite non-kooky axiom system.
You would be amazed at how many uniqueness and existence theorems in how many areas of mathematics require Zorn's Lemma. Which is, of course, equivalent to the axiom of choice. For example, "Every vector space has a (possibly infinite) basis." Or, "Every Hilbert space has a (possibly infinite) orthonormal basis."
It is rare for mathematicians to think much about choice. But it underpins key results in a surprising number of fields.
You would be amazed at how many uniqueness and existence theorems in how many areas of mathematics require Zorn's Lemma. Which is, of course, equivalent to the axiom of choice. For example, "Every vector space has a (possibly infinite) basis." Or, "Every Hilbert space has a (possibly infinite) orthonormal basis."
It is rare for mathematicians to think much about choice. But it underpins key results in a surprising number of fields.