 The Cantor theorem of the usual set theory asserts that \(|A| \lt |\wp(A)|\). This is clearly not true in NFU, since | \(V|\) is the cardinality of the universe and \(|\wp(V)|\) is the cardinality of the set of sets, and in fact \(|V| \gt \gt |\wp(V)|\) in all known models of NFU (there are many intervening cardinals in all such models). But \(|A| \lt |\wp(A)|\) does not make sense in TST: it is ill-typed. The correct theorem in TST, which is inherited by NFU, is \(|\wp_1 (A)| \lt |\wp(A)|\), where \(\wp_1 (A)\) is the set of one-element subsets of \(A\), which is at the same type as the power set of \(A\). So we have \(|\wp_1 (V)| \lt |\wp(V)|\): there are more sets than there are singleton sets. The apparent bijection \(x \mapsto \{x\}\) between \(\wp_1 (V)\) and \(V\) cannot be a set (and there is no reason to expect it to be a set, since it has an unstratified definition).