How to put equations inside brackets and define their labels?

Question

I am trying to group equations and reference them inside brackets, I tried using \label but it doesn't work, I tried split:

\[
\begin{cases} 
    \begin{split}
        \prob_1 \Big[\mathrm{Inf}_{A,1}^{+}(h^*) > \frac{1}{2} + \frac{\epsilon}{2} \Big] \leq \exp{\Big(- \frac{d \epsilon^2}{18}\Big)}
        \label{res1plus1}
    \end{split} \\
    \begin{split}
        \prob_1 \Big[\mathrm{Inf}_{A,0}^{-}(h^*) > \frac{1}{2} + \frac{\epsilon}{2} \Big] \leq \exp{\Big(- \frac{d \epsilon^2}{18}\Big)}
        \label{res1minus0}
    \end{split} \\
    \begin{split}
        \prob_1 \Big[\mathrm{Inf}_{A,0}^{+}(h^*) > \frac{1}{2} - \frac{\epsilon}{2} \Big] \leq \exp{\Big(- \frac{d \epsilon^2}{2}\Big)}
        \label{res1plus0}
    \end{split} \\
    \begin{split}
        \prob_1 \Big[\mathrm{Inf}_{A,1}^{-}(h^*) > \frac{1}{2} - \frac{\epsilon}{2} \Big] \leq \exp{\Big(- \frac{d \epsilon^2}{2}\Big)}
        \label{res1minus1}
    \end{split}
\end{cases}
\]

However, I get what i want but with no reference, as shown here:

Answer 1

I guess you want empheq. However:

1. \Big should be either \Bigl or \Bigr;
2. \exp doesn't take an argument;
3. \mathrm{Inf} should be an operator;
4. split was, in any case, useless.

\documentclass{article}
\usepackage{amsmath,amssymb,empheq}

\DeclareMathOperator{\prob}{\mathbb{P}}
\DeclareMathOperator{\Inf}{\Inf}

\begin{document}

\begin{empheq}[left=\empheqlbrace]{align}
& \prob_1 \Bigl[\Inf_{A,1}^{+}(h^*) > \frac{1}{2} + \frac{\epsilon}{2} \Bigr]
  \leq \exp\Bigl(- \frac{d \epsilon^2}{18}\Bigr)
  \label{res1plus1}
\\[0.5ex]
& \prob_1 \Bigl[\Inf_{A,0}^{-}(h^*) > \frac{1}{2} + \frac{\epsilon}{2} \Bigr]
  \leq \exp\Bigl(- \frac{d \epsilon^2}{18}\Bigr)
  \label{res1minus0}
\\[0.5ex]
& \prob_1 \Bigl[\Inf_{A,0}^{+}(h^*) > \frac{1}{2} - \frac{\epsilon}{2} \Bigr]
  \leq \exp\Bigl(- \frac{d \epsilon^2}{2}\Bigr)
  \label{res1plus0}
\\[0.5ex]
& \prob_1 \Bigl[\Inf_{A,1}^{-}(h^*) > \frac{1}{2} - \frac{\epsilon}{2} \Bigr]
  \leq \exp\Bigl(- \frac{d \epsilon^2}{2}\Bigr)
  \label{res1minus1}
\end{empheq}

\end{document}