Proof of Choice Coherence in Kreps (2013) In the first chapter of Kreps (2013), there is a proof that the choice function satisfies choice coherence. Kreps writes:



I do not understand how the third sentence of (b) logically follows from the previous two sentences. How do we necessarily know that there is a third commodity, z, in A? There might just be two commodities. Furthermore, how do we necessarily know that u(z) > u(y)? Why can't the consumer possibly be indifferent between z and y?
If there is no $z \in A$ such that
$u(z) &gt; u(y)$ (perhaps because no other $z$ exists, perhaps for other reasons)
then by definition of $c_u$ we know that $y \in c_u(A)$. This would contradict the assumption of the third sentence, that $y \notin c_u(A)$.


  If, moreover, $y \notin c_u(A)$...