\documentclass[11pt]{amsart}
\usepackage{lmodern}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{microtype}
\pagestyle{empty}
\usepackage{mathtools}
\DeclarePairedDelimiterX\setA[2]\lbrace\rbrace{\,#1\suchthatB#2\,}
\newcommand\suchthatA{\,\mathclose{}{:}\mathopen{}\,}
\DeclarePairedDelimiterX\setB[2]\lbrace\rbrace{\,#1\suchthatA#2\,}
\newcommand\suchthatB{\:\mathclose{}{:}\mathopen{}\:}
\DeclarePairedDelimiterX\setC[2]\lbrace\rbrace{\,#1\suchthatB#2\,}
\newcommand\suchthatC{\ \mathclose{}{:}\mathopen{}\ }