The "there is at least one class which.." doesn't mean "there is at least one class in that class of mutually exclusive classes which..."; it means "there exist

The "there is at least one class which.." doesn't mean "there is at least one class in that class of mutually exclusive classes which..."; it means "there exists at least one class at all which...". This is one way of phrasing the axiom of choice (although Russell uses "class" instead of "set"; in modern-day usage, we distinguish classes from sets, and AC applies to sets only).