Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-empty intersection with S. We define a binary relation R on S by , which is entire by assumption.
Let S be a set that contradicts the axiom of regularity, meaning every element in S has an intersection with itself that is not empty. We define a binary relation R on S based on the assumption that it is entire.