For sets X and Y, if |X| ≤ |Y| and |Y| ≤ |X|, then |X| = |Y|. We write |X| ≤ |Y| if there is a one-to-one map from X to Y. So the theorem states that if there is an injection from X to Y and an injection from Y to X, then there is a bijection between X and Y. The theorem proves that the relation ≤ for cardinal numbers is anti-symmetric.