Weird formatting in a smaller created empty set symbol

Question

A while back I created my own sort of emptyset symbol because the "not equal" slash didn't line up and it annoyed me. So I defined my own emptyset via \DeclareMathOperator{\vempty}{{\centernot{\Circle}}}. However, when I try to use it in a subscript (as demonstrated below), the slash becomes unaligned with the circle. Does anyone know how to remedy this? Here's my MWE:

\documentclass[12pt]{letter}
\usepackage{amsmath,amssymb,amsthm,centernot,cases}
\usepackage{wasysym}

\DeclareMathOperator{\vempty}{{\centernot{\Circle}}}
\DeclareMathOperator{\vempty}{{\text{\O}}}

\begin{document}
Example of difference between emptyset, varnothing, and vempty:
    \begin{equation*}
        \neq \emptyset \quad \neq \varnothing \quad \neq \vempty
    \end{equation*}

Example of pathological behavior:
    \begin{equation*}
        \int_{\varnothing} f dm \quad \int_{\vempty} f dm
    \end{equation*}
\end{document}

Answer 1

Symbols from wasysym are defined as

\mbox{\wasyfamily\symbol{<number>}}

so they don't change size in subscripts. Add \text around it.

Oh, and it's not an operator, but an ordinary symbol.

\documentclass[12pt]{letter}
\usepackage{amsmath,amssymb,amsthm,centernot,cases}
\usepackage{wasysym}

\newcommand{\vempty}{{\centernot{\text{\Circle}}}}

\begin{document}

Example of difference between emptyset, varnothing, and vempty:
\begin{equation*}
\neq \emptyset \quad \neq \varnothing \quad \neq \vempty
\end{equation*}

It's horrible also in subscripts \texttt{;-)}
\begin{equation*}
\int_{\varnothing} f dm \quad \int_{\vempty} f\,dm
\end{equation*}

\end{document}

Personal note: I consider \varnothing a symbol good for plumbers. Your variation is even uglier. ;-)