Linebreak inside \text inside equation

Question

I have text inside an equation that is too long for one line:

\documentclass{amsart}
\newcommand{\sset}[1]{\{ #1 \}}

\begin{document}
    We have $(x, y, z) \in \sset{(a, b, c) \in \mathbb{N}^3 \mid \text{they satisfy a very complicated condition that can only be described in text}}$.
\end{document}

This renders like in the picture.

As you can see, the text goes outside the boundaries of my document, which is bad. Instead, I want it to look something like this: I have produced this with

\documentclass{amsart}

\begin{document}
    We have $(x, y, z) \in \{(a, b, c) \in \mathbb{N}^3 \mid \text{they satisfy a very complicated condition that}$
    $\text{ can only be described in text}\}$.
\end{document}

This looks like really bad style. (For instance, what happens when I add a sentence in front? Then I have to rearrange the split. Also I can not use my custom command \sset here, but that's OK I guess.)

I have tried to use \linebreak or \allowbreak inside the \text, but they seem to do nothing.

What I would like to have is a command that I can use inside \text that allows the text to break at that point. Then I can put this command at multiple positions and LaTeX will decide where to put the break.

An alternative would be a command \breaktext for breakable text which I can use instead of \text.

Answer 1

Exit from math mode. But consider that your readers will have a very hard time in looking for the closing brace.

\documentclass{amsart}

\begin{document}

\section{Bad}

We have $(x, y, z) \in \{(a, b, c) \in \mathbb{N}^3 \mid$ they satisfy a
very complicated condition that can only be described in text$\}$.

\section{Good}

We have $(x, y, z)$ in the set of all triples $(a, b, c) \in \mathbb{N}^3$,
where $(x,y,z)$ satisfies a very complicated condition that can only be
described in text.

\end{document}