Axiom of foundation

Every nonempty set $A$ has an element $x\in A$ such that $x\cap A=\varnothing$.

The principle of epsilon induction says that every progressive class is universal: ifthen $\phi (x)$ holds for every set $x$. Over the other axioms of ZF, it is equivalent to the Axiom of foundation.

Given a definable operation $G(x,h)$ on a set $x$ and a function $h$ with domain $x$, there is a unique class function $F$ satisfyingfor every set $x$. Existence and uniqueness follow by epsilon induction, because the value at $x$ depends only on values at members of $x$.