Prove: $P(\mathbb{N}\setminus \left \{ 6 \right \})$~$P(\mathbb{N})\setminus P(\left \{ 6 \right \})$
I want to prove it while finding two injective function, I have found a function to prove that:
$|P(\mathbb{N}\setminus \left \{ 6 \right \})| \le |P(\mathbb{N})\setminus P(\left \{ 6 \right \})|$
But I struggle to find a function to the other side, to show that $|P(\mathbb{N})\setminus P(\left \{ 6 \right \})| \le |P(\mathbb{N}\setminus \left \{ 6 \right \})|$.
The best I did is finding this: $$g(A)=\left\{\begin{matrix} \emptyset & A=\left \{ 0,6 \right \} \\ A & 6 \notin A \end{matrix}\right.$$
Any help?
Thanks!