 The contradiction is obtained by assuming that the universe includes everything, i.e., variables range over both objects and functions, and that there are at least two distinct objects. Then one introduces a new operation (universal application in our sense): \(xy\) is the result of applying \(x\) to \(y\). Given two distinct objects 0 and 1, and on the assumption that the universe is closed under arbitrary definition by cases, there exists an object \(f\) such that \(fx = 0\), if \(xx \ne 0\), and \(fx = 1\), if \(xx = 0\). Then one chooses \(x = f\) (as \(x\) ranges over everything) and easily derives a contradiction.