Number of Primes is Infinite/Proof 5

Proof
there exist $n$ prime numbers.

Consider the Fermat number $F_n$.

From Goldbach's Theorem, $F_n$ is coprime to each of $F_0$ to $F_{n - 1}$.

Therefore there must be a prime number which is a divisor of $F_n$ which is not a divisor of any of $F_0$ to $F_n$.

But, again from Goldbach's Theorem, each of $F_0$ to $F_{n - 1}$ is coprime to every other Fermat number.

So all the $n$ prime numbers must have been exhausted being used as prime factors of $F_0$ to $F_{n - 1}$.

So there must be more prime numbers than $n$.

The result follows by Proof by Contradiction.