One often talks of the cardinality of a model (or more exactly, the cardinality of the universe of the model), such as when one is referring to countable models. Since the universe is, in this model, not a set to which one can assign cardinality, presumably this is referring to the cardinality of this universe when it is embedded in another universe in which the erstwhile proper class becomes a set. So far, so good?
Then one says that a class that has the same cardinality with the universe must be a proper class. For example, the class of ordinals. However, what does one do about the infinite set in a countable model of ZF? The universe and the set are both countable, yet the infinite set is not a proper class.