Subtle Cardinal - Theorem

Theorem

There is a subtle cardinal ≤κ if and only if every transitive set S of cardinality κ contains x and y such that x is a proper subset of y and x ≠ Ø and x ≠ {Ø}. An infinite ordinal κ is subtle if and only if for every λ<κ, every transitive set S of cardinality κ includes a chain (under inclusion) of order type λ.