By the definition of predecessor, there exists a concept Q and an object c such that Qc holds, 0 equals the number of Q, and n equals the number of objects satisfying Q that are not identical to c

Proof: Assume, for reductio, that some number, say \(n\), is such that \(\mathit{Precedes}(n,0)\). Then, by the definition of predecessor, it follows that there is a concept, say \(Q\) and an object, say \(c\), such that \(Qc \amp 0\eqclose \#Q \amp n\eqclose \#[\lambda z\, Qz \amp z\neq c]\). But by the Lemma Concerning Zero (above), \(0 = \#Q\) implies \(\neg\exists xQx\), which contradicts the fact that \(Qc\).