I need to use a symbol like this |---| with a continuous line between the two vertical ones

You can join \vdash and \dashv:

\documentclass{article}

\newcommand{\vdashv}{\vdash\joinrel\dashv}

\begin{document}

$A \vdashv B$

\end{document}

If you need a shorter symbol,

\documentclass{article}

\newcommand{\vdashv}{%
  \vdash\mathrel{\mkern-9mu}\dashv
}

\begin{document}

$A \vdashv B$

\end{document}

This is the maximum amount of backing up. With \joinrel it is -3mu.

A different realization with shorter vertical bars:

\documentclass{article}
\usepackage{stmaryrd}

\newcommand{\connection}{\mapstochar\relbar\mapsfromchar}

\begin{document}

$A \connection B$

\end{document}

I am not totally serious...

\documentclass{standalone}

\usepackage{rotating}

\begin{document}

\rotatebox[origin=c]{180}{$\dashv$}$\!\dashv$

\end{document}

Not so good as an image export

Better in Adobe Reader

No luck in Detexify

---

Update

\reflectbox (mirroring a symbol) from the graphicx package does a better job.

\documentclass{standalone}

% \rotatebox
\usepackage{rotating}

% \reflectbox
\usepackage{graphicx}

\begin{document}

\texttt{\textbackslash rotatebox} \rotatebox[origin=c]{180}{$\dashv$}$\!\dashv$
\texttt{\textbackslash reflectbox} \reflectbox{$\dashv$}$\!\dashv$

\end{document}