“For a complete purification” is a sixth qualification, and “from ritual impurity” is a seventh. Once these seven qualifications are fulfilled, the ruling applies: if a man purifies himself with it from ritual impurity, his ritual impurity is not lifted, even though the water itself is purifying.
For example, a woman has an amount of water measuring one and a half qullahs—which is considered a small quantity in the technical terminology—and she uses it privately in the bathhouse, performing a complete ablution from it. She then leaves, and a man comes after her to perform ablution with it. We say to him: “It does not lift your ritual impurity.”
The evidence is the Prophet’s prohibition of a man bathing with the leftover water of a woman, and of a woman bathing with the leftover water of a man (1). Ablution was included under the same ruling.
Thus, the Prophet prohibited performing ablution with it, and prohibition entails invalidity. Therefore, if he performs ablution with it, he has performed an act of worship in a prohibited manner, and it is consequently invalid.
One of the oddities in scholarship is that they cited this as evidence that a man may not perform ablution with a woman’s leftover water, yet they did not cite it as evidence that a woman may not perform ablution with a man’s leftover water (2). They said that it is permissible for a woman to bathe with a man’s leftover water. Yet since the evidence is one, the ruling is one, and the hadith is divided into two parts, why do we accept one part and not accept the second, despite knowing