Consider the following statement: $\forall m \in \mathbb{N}: (m>1 \wedge m|n) \Rightarrow m=n$. What kind of element is n? This is my reasoning, is it ok?
$ (m>1 \wedge m|n) \Rightarrow m=n \Leftrightarrow m \le 1 \vee m \not | n \vee m=n \Leftrightarrow False \vee m \not | n \vee m=n \Leftrightarrow m \not | n \vee m = n$. Now, suppose $m \neq n \Rightarrow m \not | n$. So $n$ is not divisible by every natural number, except itself. So $n$ is a prime number.