Point at which Prime-Counting Function becomes less than Eulerian Logarithmic Integral/Not using Riemann Hypothesis

Theorem
Let $\pi \left({n}\right)$ denote the prime-counting function.

Let $a \uparrow b$ be interpreted as Knuth notation for $a^b$.

Suppose the Riemann Hypothesis does not hold.

Then:


 * $\exists n < 10 \uparrow \left({10 \uparrow \left({10 \uparrow 964}\right)}\right): \pi \left({n}\right) - \displaystyle \int_2^n \frac {\mathrm d x} {\ln x} < 0$