A line for the infinity of prime numbers

There are numerous proofs of the infinity of prime numbers - the well-known theorem of Euclid from the Book of Elements is not missing in any basic lecture on number theory. 2015 Sam Northshield published in the American Mathematical Monthly (issue 122) a no less elegant proof of contradiction in the form of a one-liner, which I do not want to withhold from you (with short comments).

$$0 < \prod_{p} \sin \underbrace{ \left( \frac{\pi}{p} \right) }_{ < \pi, \text{ da } p > 1 } = \prod_{p} \sin \left( \frac{\pi}{p} + 2 \pi \underbrace{ \frac{ \prod_{p'} p' }{p} }_{ \in \mathbb{N} } \right) = \prod_{p} \sin \left( \pi \underbrace{ \frac{ 1 + 2 \prod_{p'} p' }{p} }_{ \in \mathbb{N}, \text{ da } \left( 1 + 2 \prod_{p'} p' \right) \notin \mathbb{P} } \right) = 0$$