Praise be to Allah.
The question may be summed up by saying that Aaminah breastfed both Faatimah and Umm Kalthoom, so Faatimah and Umm Kalthoom are sisters through breastfeeding. So it is not permissible for any of Faatimah’s sons to marry Umm Kalthoom, because she is their maternal aunt through breastfeeding. The Prophet (peace and blessings of Allaah be upon him) said: “What becomes mahram (forbidden for marriage) through breastfeeding is that which become mahram through blood ties.”Narrated by al-Bukhaari, 2645; Muslim, 1445.
The maternal aunt through blood ties is a mahram just as is the maternal aunt through breastfeeding.
Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (7/87):
Every woman who is a mahram (forbidden for marriage) because of blood ties, her equivalent through breastfeeding is also a mahram, namely mothers, daughters, sisters, paternal aunts, maternal aunts, brother’s daughters and sister’s daughters… because the Prophet (peace and blessings of Allaah be upon him) said: “What becomes mahram (forbidden for marriage) through breastfeeding is that which become mahram through blood ties.” Agreed upon. And we do not know of any differing opinion on this matter.
Being a mahram because of breastfeeding depends on two things:
1– That the child be breastfed five times, meaning that the infant grasps the nipple and … then lets go.
2– These five breastfeedings should have taken place before he reaches the age of two.
And Allaah knows best.