Removing strange vertical space after first (customized) \item

Question

I have a custom environment properties (that is almost ready for release to Github) that works like so:

\begin{properties}
\item[no max in reals] \forall a \in \Re, \exists b \in \Re : a < b
\item[minimum natural] \exists n \in \Na : \forall m \in \Na, n \le m
\end{properties}

and would produce (sans typo)

However, for a more complicated example, the spacing seems to be off:

I think this has something to do with the behavior of \NewDocumentEnvironment that I'm not accounting for…

MWE

\documentclass{article}
\usepackage{amsmath,amsthm} % \text{...}; \newtheorem{...}{...}
\usepackage{xparse,expl3}
\usepackage[hidelinks]{hyperref} % For some reason, this affects
                                 % numbering of the equations.
\ExplSyntaxOn

% makes sure math content is centered
\cs_new:Nn \property_do_fill:
 { \hskip \textwidth minus \textwidth }

% enters math mode and starts a property
% does not typeset a math comment
\cs_new:Nn \property_do_begin: 
{
  \equation
  \quad\bullet
  \property_do_fill:
}

% enters math mode and starts a property
% does typeset math comment
\cs_new:Nn \property_do_begin:n 
{
  \equation
  \quad\bullet\enspace\text{\slshape #1}
  \property_do_fill:
}

% ends this property and exists math mode
\cs_new:Nn \property_do_end:
{
  \property_do_fill:
  \endequation
}

\bool_new:N \g_property_first_item_bool

\NewDocumentEnvironment { properties } { }
 {
  \bool_set_true:N \g_property_first_item_bool
  % Delimits properties
  % starts off by beginning a property, and then redefines itself
  % to end a previous property before starting a new one.
  \DeclareDocumentCommand \item { o }
   {
    \bool_if:NTF \g_property_first_item_bool
     { \bool_set_false:N \g_property_first_item_bool }
     { \property_do_end: }
    \IfValueTF { ##1 }
     { \property_do_begin:n { ##1 } }
     { \property_do_begin: }
   }
 }
 { \property_do_end: }


\theoremstyle{definition}
\newtheorem{definition}{Definition}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Extra %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% This stuff isn't necessary for compilation, but is stuff that I use
% in the example.  Feel free, of course, to use it in your own stuff
% (LPPL and all that jazz).  I find it helps further logicall organize
% stuff, although it acts a little weird with AUCTeX's fold mode.



% Save the old \forall
\cs_set_eq:NN \for_all: \forall

% create a *new* forall that takes two arguments: the predicate and
% the other thing.  Because I forgot FOM.
\cs_new:Nn \for_all:nn
 {
  \for_all: #1,\; #2
 }
\DeclareDocumentCommand \forall { s }
 {
  \IfBooleanTF { #1 }
   \for_all:
   \for_all:nn
 }

% Save the old \exists
\cs_set_eq:NN \exists: \exists

% create a *new* forall that takes two arguments: the predicate and
% the other thing.  Because I forgot FOM.
\cs_new:Nn \exists:nn
 {
  \exists:\,#1:#2
 }
\DeclareDocumentCommand \exists { s }
 {
  \IfBooleanTF { #1 }
   \exists:
   \exists:nn
 }

\ExplSyntaxOff

\def\Na{\mathbf N}

\begin{document}
\begin{definition}[group]
  A group is a set $G$ together
    with a binary operation $*$ on $G$
    that satisfies the following axioms:
  \begin{properties}
  \item[associativity] \label{def:group:assoc}
    \forall{a,b,c \in G}{(a * b) * c = a * (b * c)}
  \item[identity] \label{def:group:identity}
    \exists{e \in G}{\forall{a \in G}{a * e = a = e * a}}
  \item[inverse] \label{def:group:inverse}
    \forall{x \in G}{\exists{b \in G}{a * b = b * a = e}}
  \end{properties}
\end{definition}
\end{document}

Answer 1

I think this is due to your code uses

\equation...\endequation

sequentially such that under the first item we have a \belowdisplayskip and for all other items we have \belowdisplayshortskip. So if we add

\belowdisplayskip\belowdisplayshortskip\relax

inside the new environment then the problem will be, at least temporarily, fixed.

Result:

More:

Ideally one should consider using align or gather for multiple equations, so maybe with something like:

\documentclass{article}
\usepackage{amsmath,amsthm} % \text{...}; \newtheorem{...}{...}
\usepackage{xparse,expl3}
\usepackage[hidelinks]{hyperref} % For some reason, this affects
                                 % numbering of the equations.
\ExplSyntaxOn

\cs_new:Nn \property_do_begin: 
{
  &\quad\bullet&
}

\cs_new:Nn \property_do_begin:n 
{
  &\quad\bullet\enspace\text{\slshape #1}&
}

\bool_new:N \g_property_first_item_bool
\DeclareDocumentCommand \item { o }
   {
    \IfValueTF { #1 }
     { \property_do_begin:n { #1 } }
     { \property_do_begin: }
   }

\theoremstyle{definition}
\newtheorem{definition}{Definition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Extra %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% This stuff isn't necessary for compilation, but is stuff that I use
% in the example.  Feel free, of course, to use it in your own stuff
% (LPPL and all that jazz).  I find it helps further logicall organize
% stuff, although it acts a little weird with AUCTeX's fold mode.



% Save the old \forall
\cs_set_eq:NN \for_all: \forall

% create a *new* forall that takes two arguments: the predicate and
% the other thing.  Because I forgot FOM.
\cs_new:Nn \for_all:nn
 {
  \for_all: #1,\; #2
 }
\DeclareDocumentCommand \forall { s }
 {
  \IfBooleanTF { #1 }
   \for_all:
   \for_all:nn
 }

% Save the old \exists
\cs_set_eq:NN \exists: \exists

% create a *new* forall that takes two arguments: the predicate and
% the other thing.  Because I forgot FOM.
\cs_new:Nn \exists:nn
 {
  \exists:\,#1:#2
 }
\DeclareDocumentCommand \exists { s }
 {
  \IfBooleanTF { #1 }
   \exists:
   \exists:nn
 }

\ExplSyntaxOff

\def\Na{\mathbf N}

\begin{document}
\begin{definition}[group]
  A group is a set $G$ together
    with a binary operation $*$ on $G$
    that satisfies the following axioms:
  \begin{align}
  \item[associativity] \label{def:group:assoc}
    \forall{a,b,c \in G}{(a * b) * c = a * (b * c)}\\
  \item[identity] \label{def:group:identity}
    \exists{e \in G}{\forall{a \in G}{a * e = a = e * a}}\\
  \item[inverse] \label{def:group:inverse}
    \forall{x \in G}{\exists{b \in G}{a * b = b * a = e}}
  \end{align}
\end{definition}
\end{document}

Check out the result: