Variant 4.7.4.21. For every pair of integers $n,d \geq 0$, there are only finitely many isomorphism classes of simplicial sets $X$ having dimension $\leq d$ and exactly $n$ nondegenerate simplices. This follows by induction on $n$, since for $n > 0$ every such simplicial set fits into a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{d'} \ar [r] \ar [d] & X' \ar [d] \\ \Delta ^{d'} \ar [r] & X } \]
for some $d' \leq d$ (see Proposition 1.1.4.12), where $X'$ is a simplicial subset having $n-1$ nondegenerate simplices. Taking the union over $n$ and $d$, we conclude that the collection of isomorphism classes of finite simplicial sets is countable.