$ \newcommand{\exi}{\exists\,} \newcommand{\all}{\forall} \newcommand{\equ}{\!=\!} \newcommand{\nequ}{\!\neq\!} \newcommand{\amp}{\;\&\;} \renewcommand{\Set}[2]{\left\{\;#1\mathrel{}\middle|\mathrel{}#2\;\right\}} \newcommand{\parenth}[1]{\left(\;#1\;\right)} \newcommand{\braces}[1]{\left\{\;#1\;\right\}} \newcommand{\bracket}[1]{\left[\;#1\;\right]} \newcommand{\godel}[1]{\left\ulcorner #1 \right\urcorner}$