Proof

Assume, there are only finitely many prime numbers $p_1, p_2,\ldots,p_r$.
Calculate the number $n:=p_1\cdot p_2\cdot \cdots \cdot p_r+1.$
By assumption $n$ is composite.
However, note that $n$ cannot be divisible by any of the primes $p_1, p_2,\ldots,p_r$, otherwise it would follow from the divisibility law no. 6 that also 1 is divisible by this prime number, which is impossible.
Therefore, there has to exist another prime dividing $n$, in contradiction to our assumption.
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