ZFC proves that every analytic set, and therefore every co-analytic set, is Lebesgue measurable and has the Baire property. It also proves that every analytic set has the perfect set property. But the perfect set property for co-analytic sets implies that the first uncountable cardinal, \(\aleph_1\), is a large cardinal in the constructible universe \(L\) (see Section 7), namely a so-called inaccessible cardinal (see  Section 10),  which implies that one cannot prove in ZFC that every co-analytic set has the perfect set property.