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I have an enumerate environment with 3 items in it, say H1, H2, and H3. I would like to reference this list as a whole, calling it H. Ideally, I could make an enumerate environment like cases and have H followed by a large left curly brace and then the enumerate environment for H1, H2, and H3. Is there a way to do this?

   _  
  /   H1
  |   
H<    H2
  |   
   \_ H3

Something like this, while being able to reference the whole list as \ref{H}.

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You can use cases inside enumitem. To be able to reference the items of the enumitem you have to create a new float to wrap the whole thing in, in my example enucase, and put a label on the \item you want to refer to. This is a rough and ready solution but works. Just note that if you want to put normal text into the items following the curly bracket you have to wrap it \text{} or something similar.

\documentclass{article}

\usepackage{amsmath} % for cases and \text{}
\usepackage[shortlabels]{enumitem} % to define the Alphabetic enumerate label

\usepackage{float} % to create the new float
\floatstyle{plain}
\newfloat{enucase}{tbhp}{loe}[section] %{float-name}{placement}{toc-style-list}[section reference]
\floatname{enucase}{Enumerated Case} %{float-name}{name-in-loe/toc}

\begin{document}

\section{Enumerated Cases with numbers}

\begin{enucase}
            \begin{enumerate}
            \item A
            \item\label{H} $ H =
            \begin{cases}
                one \\
                two \\
                three
            \end{cases}$
             \item B
            \end{enumerate}
\end{enucase}

    refers to Case \ref{H}
\section{Enumerated cases only with letters}

\begin{enucase}
            \begin{enumerate}[label=\Alph*]
            \item bla bla bla
            \item\label{B} $ =
            \begin{cases}
                \text{item one} \\
                \text{item two} \\
                \text{item three}
            \end{cases}$
             \item bla bla
            \end{enumerate}
\end{enucase}

refers to Case \ref{B}

\end{document}

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