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.
Consider a set S that violates the axiom of regularity; this means each element within S intersects with itself, creating a non-empty intersection.  We establish a binary relationship R on S using the definition of "entire" as our basis.