Euclid's Theorem

I came across this beautiful proof in GEB and wanted to reproduce it here.

Proof that there are infinitely many primes:

Assume there are finitely many primes. Let’s denote the finite set of all primes by $P=\{p_1,p_2,...,p_n\}$.

Consider the number $x=1+\prod _{i=1}^n{p}_i$.

$x$ has a remainder of $1$ for all primes $p_i$ in $P$ (and therefore is not divisible by any of them). Therefore, the prime factorization of $x$ includes primes not in the set $P$.

However, we stated that $P$ was the set of all primes. Finding a prime outside of that set is a contraction of our assumption. Therefore, our assumption is false.