[Tex/LaTex] Multi-column delimiter in align

Question

I want to set the following, but with all the <'s aligned horizontally

Source of this failed attempt:

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}

\newcommand{\ra}{\rightarrow}
\newcommand{\T}[1]{\mathbb{T}_{#1}}
\newcommand{\lbe}{<_{\beta\eta}}

\begin{align*}
0 & \lbe 0 \ra 0                && \in \T0 \\
&\left.
\begin{aligned}
& \lbe 0^2 \ra 0 \\
& \lbe \cdots \\
& \lbe 0^k \ra 0 \\
& \lbe \cdots
\end{aligned}
\right\}                        && \in \T1 \\
& \lbe 1 \ra 0 \ra 0            && \in \T2 \\
& \lbe 1 \ra 1 \ra 0 \ra 0      && \in \T3 \\
& \lbe 3 \ra 0 \ra 0            && \in \T4 \\
& \lbe 1_2 \ra 0 \ra 0          && \in \T5
\end{align*}

\end{document}

Answer 1

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}

\newcommand{\ra}{\rightarrow}
\newcommand{\T}[1]{\mathbb{T}_{#1}}
\newcommand{\lbe}{<_{\beta\eta}}

\begin{align*}
0 & \lbe 0 \ra 0                && \in \T0 \\
&\left.
\renewcommand\arraystretch{1.3}
\begin{array}{@{}l}
\lbe\displaystyle 0^2 \ra 0 \\
\lbe\displaystyle \cdots \\
\lbe\displaystyle 0^k \ra 0 \\
\lbe\displaystyle \cdots
\end{array}
\right\}                        && \in \T1 \\
& \lbe 1 \ra 0 \ra 0            && \in \T2 \\
& \lbe 1 \ra 1 \ra 0 \ra 0      && \in \T3 \\
& \lbe 3 \ra 0 \ra 0            && \in \T4 \\
& \lbe 1_2 \ra 0 \ra 0          && \in \T5
\end{align*}

\end{document}

This is basically what the cases environment does except that it has two columns, the brace is on the other side and the \arraystretch is set to 1.2.

Edit:
This is basically how aligned works except that I removed the \null\,.

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}

\makeatletter
\newenvironment{alignedleft}[1][c]{%
    \RIfM@\else
        \nonmatherr@{\begin{\@currenvir}}%
    \fi
    \if #1t\vtop \else \if#1b \vbox \else \vcenter \fi \fi \bgroup
        \def\math@cr@@@{\cr}%
        \Let@ \chardef\dspbrk@context\@ne
        \default@tag
        \spread@equation % no-op if already called
        \ialign\bgroup
           &\strut@
            $\m@th\displaystyle{##}$%
            \hfil
            \tabskip\minalignsep
            \crcr
}{%
    \crcr
    \egroup
    \egroup
}
\makeatother

\begin{document}

\newcommand{\ra}{\rightarrow}
\newcommand{\T}[1]{\mathbb{T}_{#1}}
\newcommand{\lbe}{<_{\beta\eta}}

\begin{align*}
0 & \lbe 0 \ra 0                && \in \T0 \\
&
\def\arraystretch{1.206}
\left.
\begin{alignedleft}
\lbe 0^2 \ra 0 \\
\lbe \cdots \\
\lbe 0^k \ra 0 \\
\lbe \cdots
\end{alignedleft}
\right\}                        && \in \T1 \\
& \lbe 1 \ra 0 \ra 0            && \in \T2 \\
& \lbe 1 \ra 1 \ra 0 \ra 0      && \in \T3 \\
& \lbe 3 \ra 0 \ra 0            && \in \T4 \\
& \lbe 1_2 \ra 0 \ra 0          && \in \T5
\end{align*}

\end{document}