Can an artificer cast spells using a single component from a set of artisan’s tools?

Under Tools Required

You produce your artificer spell effects through your tools. You must have a spellcasting focus – specifically tinker’ tools or some kind of artisan’s tool – in hand when you cast any spell with this Spellcasting feature. You must be proficient with the tool to use it in this way.

"Artisan’s tool" is singular. But it can refer to a set of objects.

For example:

Weaver’s tools include thread, needles, and scraps of cloth.

Weaver’s tools are an artisan’s tool but does an artificer proficient with them need to hold the needle, thread, and cloth in one hand to cast spells?

For example, the party has been striped of all their belongings but not their clothes. Can an artificer with proficiency in weaver’s tools tear off a scrap of their clothing and begin casting?