This is a ridiculous lemma that I came up with.
Lemma. Let be a (commutative) ring, and let be its total ring of fractions. Then and have the same cardinality.
Proof. If is finite, my previous post shows that . If is infinite, then is a subquotient of , hence . But injects into , so .
Corollary. If is a domain, then .
Proof. This is a special case of the lemma.