Prove there exists a countably infinite subset $A$ of $P(\mathbb{N})$ that satisfies given conditions Prove that there exists a countably infinite set $A \subseteq $ $P(\mathbb{N})$ that satisfies all of the following conditions:
$i)$ $X \cap  Y = \emptyset$ for all $X, Y \in A$ such that $X \neq Y$
$ii)$ $\mathbb{N} = \bigcup A $
$iii)$ Every element in $A$ is countably infinite 
For all the sets I tried, the three conditions were satisfied but $A$ was not countably infinite, so now I'm stuck. Any help would be greatly appreciated.

Define the set
$$
A_1=\{1\}\cup\{n\in\mathbb{N}:n\;\mathrm{is\; product\; of\; two\; or \; more\; distinct \; primes}\}
$$
And for each prime number $p$, let $A_p\{p^n:n\in\mathbb{N}$. Finally, let $A$ be the collection of this sets. It is not hard to see that this $A$ is the desires partition of $\mathbb{N}$.
