In the proof of lemma 6.7 of 'A Course in Mathematical Logic', in page $63$ Manin writes,
" We consider the set $S$ of formulas $P(x)$ containing one free variable $x$ and such that $\neg\forall xP(x)\in \mathscr{E}\ \cup \text{Ax}L'$. For each $P(x)$ in $S$ we choose a new constant $c_P$ subject to the following restriction: each $c_P$ can be assigned a natural number, its $rank$, in such a way that if a constant of rank $n$ occurs in $P(x)$ then $c_P$ has rank $>n$. This can be done since card($S)\leq$ card(alphabet of $L')+\aleph_0=$ card(alphabet of $L)+\aleph_0$"
I am unable to see how the assignment of $c_P$ is justified.