Pandigital Prime Sets

Using all of the digits $1$ through $9$ and concatenating them freely to form decimal integers, different sets can be formed. Interestingly with the set ${2, 5, 47, 89, 631}$ , all of the elements belonging to it are prime.

How many distinct sets containing each of the digits one through nine exactly once contain only prime elements?

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.