About the existence of the diagonal set of Cantor The classic proof of the Cantor set start with the assumption that the set
$$B=\{x\in A:x\notin f(x)\}$$
exists, where $f: A\to\mathcal P(A)$ is a bijective function. I understand the proof but I dont understand the assumptions where you start to make this proof.
To be clear, why a set $B$ can be constructed? How you can justify this assumption? To me the proof of the Cantor theorem is far to be clear or complete if there are not an explanation about why $B$ must be possible.
Then, can someone explain to me or justify, via other theorems if possible, why $B$ must exist? Thank you in advance.
P.S.: can someone explain to me the downvotes in the question?

Another example, from the infinite set of natural numbers:
\begin{eqnarray}
S=\mathbb{N}\\
P\left(S\right)=\{\phi, S, \forall U\neq\phi;U\subset S\}\\
f\left(x\right)=\{x\}
\end{eqnarray}
Then f is 1-1.
$B=\{x\in S;x\notin f\left(x\right)\}$
Then
$\forall x\notin B$
$B=\phi \in P\left(S\right)$
Which exists and again is not in the image of $f$.
