In the beginning of my mathematics studies at university, we have learnt that nearly all of ordinary mathematics not dealing with proper classes can be formalized within ZFC, which is a famous axiomatization of set theory. Among these ZFC-axioms, there is the following technical assumption:
Axiom of Regularity: Every non-empty set $A$ contains an element that is disjoint from $A$.
Having this axiom available, one can easily prove that there is no infinite sequence $(a_n)_{n\in\mathbb N}$ of sets such that $$a_{i+1}\in a_i\qquad \text{for all $i\in \mathbb N$}.$$ In particular, there is no set $A$ with $A\in A$; and there are no sets $A$, $B$ such that $A\in B$ and $B\in A$ and so on. Furthermore, one can prove that every set is in some stage of the cumulative hierarchy using the axiom of regularity.
These consequences are just applications of regularity within set theory. That is why I was wondering whether there are applications of the axiom of regularity to (ordinary) non-set-theoretical mathematics. In my mathematics courses at university we have never again used the axiom of regularity.
In order to put it in a nutshell, let me repeat what my question is:
What are some applications of the axiom of regularity to non-set-theoretical mathematics? Are there some applications of the axiom of regularity to non-set-theoretical mathematics at all?
