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Proof

(related to Corollary: Prime Dividing Product of Primes Implies Prime Divisor)

• By the generalized Euclidean lemma, there is at least one divisor with $p\mid p_i.$
• Since the prime number $p_i$ has only the trivial divisors $1$ and $p_i$ and since $p\neq 1,$ it follows $p=p_i.$
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References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927