In a the poset \$$P(X) \$$, the complement of the top element \$$X \$$ defines negation,
$$\neg(X): P(X) \rightarrow P(X) \\ x \mapsto X \setminus x.$$

Proof.

Let \$$x=X \$$, then \$$X \setminus X = \varnothing \$$.
Let \$$x=\varnothing \$$, then \$$X \setminus \varnothing = X\$$.
Finally, let \$$x=y,\$$ for some \$$y, \\, \varnothing \subsetneq y \subsetneq X \$$, then there will exist some subset \$$z, \varnothing \subsetneq z \subsetneq X \$$ where \$$z \neq y \$$ and \$$z = X \setminus y \$$.