How do I strikethrough in text{} inside inline math mode

You can use the sout command from ulem.

documentclass{article}
usepackage{newpxtext,newpxmath} % These fonts are just examples.
usepackage{ulem}

begin{document}
$mathfrak{P}:={textnormal{set of all sout{prime} ideals contained in}  Asetminus S}$
end{document}

You usually want textnormal, rather than text, so that the formatting of the surrounding text will not bleed through.