documentclass{article}
usepackage{amssymb}
newcommanddoesnotentail{mkern-2munotmkern2muvdash}
begin{document}
$Adoesnotentail B$

$Avdash B$
end{document}

documentclass{article}
usepackage{amssymb,stackengine}
newcommanddoesnotentail{mathrel{ensurestackMath{%
  stackengine{.175em}{vdash}{scriptstyle/}{O}{c}{F}{F}{L}}}}
begin{document}
$Adoesnotentail B$ via stackengine

$Avdash B$ textbackslash vdash

$Anvdash B$ textbackslash nvdash
end{document}

documentclass{article}
usepackage{amsmath,graphicx}

DeclareRobustCommand{notvdash}{%
  mathrel{mathpalettevdashnotvdash}%
}
makeatletter
newcommand{vdashnot}[2]{%
  sbox0{$m@th#1#2$}%
  sbox2{raisebox{depth}{$m@th#1/$}}%
  ooalign{%
    hidewidthkern0.1emresizebox{1.1width}{ht0}{box2}hidewidthcr
    box0cr
  }%
}
makeatother

begin{document}

$Anotvdash B$

end{document}