Wedge with vertical bar

This overlaps three things: wedge/vee, shortmid, and shortmid raised to a correct height. The numbers suit cm and lm fonts, for other fonts they might need to be modified.

See here for a quick course on \ooalign.

\documentclass{article}
\usepackage{amssymb}

\makeatletter
\newcommand\@@vertwv[3]{\mathop{\ooalign{%
  $#1#3$\cr
  \hfil$#1\shortmid$\hfil\cr
  \hfil\raisebox{#2}{$#1\shortmid$}\hfil\cr}}}
\newcommand\@vertwv[1]{\mathchoice
  {\@@vertwv{\displaystyle     }{0.38ex}{#1}}%
  {\@@vertwv{\textstyle        }{0.38ex}{#1}}%
  {\@@vertwv{\scriptstyle      }{0.28ex}{#1}}%
  {\@@vertwv{\scriptscriptstyle}{0.21ex}{#1}}}
\newcommand\vertwedge{\@vertwv\wedge}
\newcommand\vertvee  {\@vertwv\vee  }
\makeatother

\begin{document}

\[ a\vertwedge b \neq a\vertvee b \]
\[ \textstyle a\vertwedge b \neq a\vertvee b \]
\[ \scriptstyle a\vertwedge b \neq a\vertvee b \]
\[ \scriptscriptstyle a\vertwedge b \neq a\vertvee b \]

\end{document}

For users of XeLaTeX and LuaLaTeX exist with the unicode-math package and the fonts XITS Math, Asana Math and (according to the unciode-math symbol list) Lucida Math the macros

• \veemidvert and
• \wedgemidvert

to access these symbols.

Code

\documentclass{article}
\usepackage{amsmath}% not needed but used for this example and the align environment
\usepackage{unicode-math}
\setmathfont{XITS Math}
%\setmathfont{Asana Math}
\begin{document}
\begin{align*}
      x \veemidvert y \leq z & \equiv x \leq z \wedge y \leq z \\
    z \leq x \wedgemidvert y & \equiv z \leq x \wedge z \leq y
\end{align*}
\end{document}

Output

The top rows show XITS Math, the bottom ones show Asana Math.