\begin{definition}\label{universal_set}
    A set $V$ is universal iff
    %every set is an element of $V$.
    for all sets $x$ we have $x\in V$.
\end{definition}

\begin{theorem}\label{no_universal_set}
    There exists no universal set.
\end{theorem}
\begin{proof}
    Suppose not.
    Take a universal set $V$.
    Let $R = \{ x\in V \mid x\not\in x \}$.
    Then $R\in R$ iff $R\not\in R$.
    Contradiction.
\end{proof}