Is this sufficient proof for the existence of a set $X$?

Question

There exists a set $X$ for which $\Bbb{N}\in X$ and $\Bbb{N}\subseteq X$.

Let $X=\Bbb{N}\cup\mathscr{P}(\Bbb{N})$

$\blacksquare$

Forgive me if this is a stupid question but is this a valid example? I know that a set is a subset of itself and that a set is an element of its power set but I don't know if this is a valid example for the given existential statement. Thanks for your time! :)

Answer 1

It is a valid choice of $X$, but you should probably explain why $\mathbb{N} \in X$ and $\mathbb{N} \subseteq X$ for that $X$.

(You could also have used $X = \mathbb{N} \cup \{\mathbb{N}\}$, which is the minimal choice that works.)

Answer 2

More simple would be $X = \{\mathbb{N}\} \cup \mathbb{N}$, but I suppose your example works.