well-foundedness and axiom of foundation
For any , the collection is a set, and
for any non-empty , there is an element such that if , then .
Given ZF, is a well-founded relation iff the Axiom of Foundation (AF) is true.
We will prove this using one of the equivalent versions of AF: for every non-empty set , there is an such that .
Suppose is well-founded and a non-empty set. We want to find such that . Since is well-founded, there is a -minimal set such that . Since no set such that and (otherwise would not be -minimal), we have that .