It is well known that there is no set of all sets. The argument goes as follows:

Define:

$$\mathcal{C}:=\{X: X\ \textrm{is a set and}\ X\not\in X\}.$$

If $\mathcal{C}$ was a set then $$\mathcal{C}\in \mathcal{C}\Leftrightarrow \mathcal{C}\not\in \mathcal{C}.$$

I am asking myself why this would show there is not a set of all sets. 

Well, I think if there was a set of all sets it should be equal to $\mathcal{C}$, right? But this would rely on the fact that every set is NOT a member of itself, but what would justify that?

I am not even sure if I understand the definition of $\mathcal{C}$ anyway.

Can anyone give me further clarifications?

Thanks.