Does a small, involuntary amount of vomit invalidate the fast when brushing teeth with a toothbrush and toothpaste?

If a fasting person did not intentionally vomit, and the vomit reached their throat then they expelled it without swallowing it, their fast is valid. However, if they swallowed it by choice after it reached a point from which it could be expelled, their fast is broken and they must make up for it. If the vomit returned to their stomach unintentionally, their fast is not broken, based on the Prophet's (peace be upon him) saying: "Whoever is overcome by vomit does not have to make up for the fast. And whoever intentionally vomits must make up for it."