Is it good representation of set of prime numbers ?

Question

$$ P = \{ x € N : \exists x \forall y \space \space x\neq y \space \land \space y\neq x \space \land \space y\nmid x \} $$

Trying to find answer for couple of hours now. Is it good representation of set of prime numbers ? (P should be that set)

Answer 1

As Lee points out in the comments above, your version is nonsensical.

There are a few different ways we could proceed. Probably the simplest is $$\Bbb P=\bigl\{x\in\Bbb N\setminus\{1\}:\forall y\in\Bbb N\setminus\{1,x\},y\nmid x\bigr\}.$$

Another way to go, assuming $0\notin\Bbb N,$ is $$\Bbb P=\bigl\{x\in\Bbb N\setminus\{1\}:\forall y\in\Bbb N,\forall z\in\Bbb N,(x\mid yz)\implies(x\mid y)\vee(x\mid z)\bigr\}.$$