Proof that the factorial is nonelementary

We will use the following facts:

(i) The extension, to a larger domain, of a non-elementary function is also non-elementary;

(ii) The derivative of an elementary function is also elementary;

(iii) The product of finitely many elementary functions is also elementary;

(iv) The product of an elementary function times a non-elementary function is non-elementary.

Claim 1: $\Gamma(x)$ is a non-elementary function.

Proof. Assume the contrary. By (i) $n!$ must be elementary, and by (ii) so is $\Gamma'(x)=\Gamma(x)\psi^{(0)}(x)$, which by (iii) implies the same for $\psi^{(0)}(x)$ and all of its derivatives. But we have $$\psi^{(n)}(x)=(-1)^{n+1}\ n!\ \zeta(n+1,x),$$ where $\zeta(a,s)$ is the non-elementary Hurwitz zeta function, so combining (iii) and (iv) yields that $\psi^{(n)}(x)$ is a non-elementary function, contradiction. $ \ \ \ \text{QED} $

Claim 2: $n!$ is a non-elementary function.

Proof. The Riemann zeta function satisfies $$\begin{align} 2\ \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)&= \int_0 ^ \infty \left(\vartheta(0,it) -1\right)t^{s/2-1}dt,\end{align}$$ where $\vartheta(z,q)$ is the non-elementary Jacobi theta function. So let $s=2n$ to obtain $$\begin{align}2\pi^{-n}\Gamma(n)\zeta(2n) &= \int_0 ^ \infty \left(\vartheta(0,it) -1\right)t^{n-1}dt \\ 2\ \pi^{-n} (n-1)! \frac{ (-1)^{n+1} B_{2n}(2\pi)^{2n}}{2(2n)!} &= \int_0 ^ \infty \left(\vartheta(0,it) -1\right)t^{n-1}dt \\ - (-\pi)^n 2^{2n} B_{2n}\frac{(n-1)!}{(2n)!}&=\int_0 ^ \infty \left(\vartheta(0,it) -1\right)t^{n-1}dt.\end{align}$$ Now, by (i) the Bernoulli numbers are elementary, due to being a restriction of the Bernoulli polynomials, which are elementary. But the RHS is non-elementary by (ii), therefore by (iii) the ratio of factorials is non-elementary as well, and the claim follows combining this and (iii). $ \ \ \ \text{QED} $

Of course Claim 1 directly follows from Claim 2 by (i), but I wanted to give two different and independent proofs.