We can say that $p$ is prime if and only if :
$$p\ne 1$$ $$p\space is\space prime \leftrightarrow \left\{(1|p)\wedge(p|p)\wedge\forall i\space2\leq i\leq p-1 (p\equiv r(mod\space i),r\neq0\right),(p,r)\in N\}$$
$$Edit: $$
$$p\ne 1$$ $$p\space is\space prime \leftrightarrow \left\{(\forall i\space2\leq i\leq p-1 (i\not\mid p)\right),(p,i)\in \mathbb{Z^{+}}\}$$