Exponentials of non-singular simplicial sets

A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set $X^K$ has $n$-simplices given by the simplicial maps $\Delta[n] \times K \to X$. We prove that $X^K$ is non-singular whenever $X$ is non-singular. It follows that non-singular simplicial sets form a cartesian closed category with all limits and colimits, but it is not a topos.