Right. But this is only a proof in a given theory. If ZFC was consistent and complete, it would have had complete dominance over the notion of truth, even if it had some quirks (as the joke goes, "The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn’s lemma?"). It is not the case, so there are competing theories with different consequences, and the world is more interesting because of that.

What I'm trying to get at is that mathematical truth people care about goes beyond provability, and that is the case in good part because of incompleteness.