Euclid's Lemma for Prime Divisors/General Result/Proof 3

Proof
Let $p \divides n$.


 * $\forall i \in \set {1, 2, \ldots, r}: p \nmid a_i$
 * $\forall i \in \set {1, 2, \ldots, r}: p \nmid a_i$

By Prime not Divisor implies Coprime:
 * $\forall i \in \set {1, 2, \ldots, r}: p \perp a_i$

By Integer Coprime to all Factors is Coprime to Whole:
 * $p \perp n$

By definition of coprime:
 * $p \nmid n$

The result follows by Proof by Contradiction.