\documentclass{article}
\usepackage{geometry}
\usepackage{bussproofs}
\geometry{landscape}
\pagestyle{empty}
\begin{document}

\begin{small}
\AxiomC{$x_{0}\mathrel{:}\exists x.(P(x) \wedge \neg P(x))$}
\AxiomC{$x_{1}\mathrel{:}P(a) \wedge \neg P(a)$}
\RightLabel{\tiny($\wedge$EL)}
\UnaryInfC{$P(a)$}
\AxiomC{$x_{1}\mathrel{:}P(a) \wedge \neg P(a)$}
\RightLabel{\tiny($\wedge$ER)}
\UnaryInfC{$\neg P(a)$}
\RightLabel{\tiny($\neg$E)}
\UnaryInfC{$P(a) \Rightarrow \bot$}
\RightLabel{\tiny($\Rightarrow$E)}
\BinaryInfC{$\bot$}
\RightLabel{\tiny($\Rightarrow$I/$x_{1}$)}
\UnaryInfC{$P(a) \wedge \neg P(a) \Rightarrow \bot$}
\RightLabel{\tiny($\exists$E)}
\BinaryInfC{$\bot$}
\RightLabel{\tiny($\Rightarrow$I/$x_{0}$)}
\UnaryInfC{$\exists x.(P(x) \wedge \neg P(x)) \Rightarrow \bot$}
\RightLabel{\tiny($\neg$I)}
\UnaryInfC{$\neg \exists x.(P(x) \wedge \neg P(x))$}
\DisplayProof
\end{small}

\end{document}
