Space that disappears at the edges (beginning/end) of a math formula

Question

Is there a way to produce a space that disappears at the beginning or end of a math formula?

Idea: For the formula-final case, something that looks ahead or takes an obligatory argument might be a start. I remember that commands whose obligatory arguments are terminated by the end of a math formula will be fed empty arguments.

Some practical examples:

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Example 1

The uniqueness quantifier ("there exists exactly one") is often written "∃!".

Out of visual considerations I would like the following variable to be a bit away from the exclamation mark, to make it clearer that "∃!" is considered a single token ("token" here in the sense of parsing in a programming language). I can define something like \newcommand*{\existsunique}{\exists!\mskip1.5mu\relax}, but the space of 1.5mu remains if the symbol is used by itself:

\documentclass{article}
\usepackage{amsmath} % better spacing behavior for \colon
\usepackage{dsfont}

\newcommand*{\existsunique}{\exists!\mskip1.5mu\relax}


\begin{document}

% spacing (1.5mu) as desired
\(\forall x \in \mathds{R}^{+}_{0} \colon
  \existsunique y \in \mathds{R}^{+}_{0} \colon
  y^2 = x\)

% spacing (0mu) too tight
\(\forall x \in \mathds{R}^{+}_{0} \colon
  \exists! y \in \mathds{R}^{+}_{0} \colon
  y^2 = x\)

% spacing (1.5mu) superfluous
The quantifier \(\existsunique\) means ``there is exactly one''.

% spacing (0mu) as desired
The quantifier \(\exists!\) means ``there is exactly one''.

\end{document}

Here the first lines of each pair have additional spacing of 1.5mu after the uniqueness quantifier "∃!", while the second lines of each pair don't. For the first pair, the added spacing is as desired; for the second pair, it doesn't look good.

In this example, I would like a 1.5mu space that disappears at the end of a formula: the same macro should produce lines 1 and 4 above. Perhaps there is a different approach to "∃!", but this serves as one practical example to my general question above.

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Example 2

For Heiko Oberdiek's intermediate-length arrows

• \Implies (between \Rightarrow and \implies in length)
• \Impliedby (between \Leftarrow and \impliedby in length)
• \Iff (between \Leftrightarrow and \iff in length)

defined here, contextual deletion of spacing is desirable on both sides.

Answer 1

This revised answer grew out of seeing egreg's answer at What is the difference of \mathop, \operatorname and \DeclareMathOperator?, where he mentioned this construct: \newcommand{\diff}{\mathop{}\!d}

In the OP's question above, Werner and egreg made suggestions using \mathbin and \mathop, but the OP replied "I also considered \mathop, but I'd like narrower spacing."

So taking a tack similar to \diff, I came up with this:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{dsfont}
\usepackage[usestackEOL]{stackengine}
\def\stackalignment{l}
\def\showall{~\fbox{\Longstack{
\text{\detokenize\expandafter{\existsunique}}\\
A \colon  \existsunique y \\
\fboxsep0pt\fbox{$\existsunique$}\\
A \colon  {\existsunique} y
}~}\par}
\begin{document}
\savestack{\Header}{\Longstack{Definition:\\within math:\\boxed:\\brace-isolated:}}
\stackMath
\Header\def\existsunique{\exists!}\showall
\Header\def\existsunique{\exists!\mathop{}\!}\showall
\Header\def\existsunique{\mathop{}\!\exists!\mathop{}\!}\showall
\end{document}

While the actual \! kerns may be different than the OP's tastes, perhaps the approach could be adapted to kerns of his choice.

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ORIGINAL APPROACH:

This solution may (or may not) give you something that works for your application. I introduce the macro \mymathop{op-name}{op-definition}{pre-kern}{post-kern}.

What it does is look at what token follows the new "mymathop", in your case, \existsunique. If what follow is a $, \), or \egroup, it prints out the "op definition" by itself; otherwise, it adds the pre- and post-kern about the "op-definition" (as a side note, the use of \@ifnextchar will swallow any intervening spaces between the "mymathop" and the next token).

The \egroup catch allows one to force \mymathop to suppress the pre-/post-\kern even in the middle of a formula, by enclosing it in braces

The MWE below shows it in a formula, fboxed by itself with two of the delimiter styles, and then inside a formula, isolated, with its own set of braces.

\documentclass{article}
\usepackage{amsmath}
\usepackage{dsfont}
\makeatletter
\def\mymathop#1#2#3#4{%
  \def#1{\@ifnextchar${#2}{\@ifnextchar\){#2}{\@ifnextchar\egroup{#2}{\kern#3#2\kern#4}}}}
}
\makeatother
\mymathop{\existsunique}{\exists!}{0pt}{3pt}
\begin{document}
\(\forall x \in \mathds{R}^{+}_{0} \colon
  \existsunique y \in \mathds{R}^{+}_{0} \colon
  y^2 = x\)

\fboxsep0pt\fbox{$\existsunique $} \fbox{\(\existsunique \)}

\(\forall x \in \mathds{R}^{+}_{0} \colon
  {\existsunique} y \in \mathds{R}^{+}_{0} \colon
  y^2 = x\)
\end{document}

Answer 2

I think

\newcommand*{\existsunique}{\thinmuskip=1.5mu\mathop{\exists!}\relax}

does what you want.

I think a better solution is

    \makeatletter
     \newcommand*{\existsuniquenew}{\@ifnextchar{\)}{\@end}{%
       \@ifnextchar{$}{\@end}{\not@end}}}
     \def\@end{\exists!\relax}%
     \def\not@end{\exists!\mskip1.5mu\relax}
    \makeatother