I want to prove "There is no set to which every function belongs." Can I approach it as follows?
Attempt No. 1: Let $$F: A\rightarrow B$$ Since $$F\subset A\times B,$$ it follows that $$F\in \mathcal{P}(A\times B).$$ Now, let $$\mathcal{P}(A\times B)$$ be the set of all functions from A into B. Since $$\mathcal{P}(A\times B)\subseteq \mathcal{PP}(A\times B),$$ it follows that $$\mathcal{PP}(A\times B)$$ is also a set of all functions from A into B. Therefore $$\mathcal{P}(A\times B)=\mathcal{PP}(A\times B).$$
Attempt No. 2 is to approach it by the concept of Russel Paradox. Then I must build a set of all functions that are not in itself, but, honestly, I have not a clue of what constitute a function that is not in itself.