In the Hadith of Thaqalayn
From the hadith of Thaqalayn, superiority of the Ahl al-Bayt (a) over others can easily be understood, because the Prophet (s) put them beside the Qur'an, called the Qur'an as "Thiql Akbar" and Ahl al-Bayt (a) as Thiql Asghar and never put another person beside the Qur'an. Therefore, the Ahl al-Bayt (a) are superior to others the same way as the noble Qur'an is superior over Muslims.
Accordingly, Sa'd al-Din Taftazani has said, "the message of the verse of Tathir and Hadith of Thaqalayn is the superiority of the Ahl al-Bayt (a) over others and the reason of their superiority is not only because of their relation with the Prophet (s), because according to the Qur'an, tradition and consensus, the criterion of superiority is knowledge and Taqwa which Ahl al-Bayt (a) have and this can be understood from their equal position with the Qur'an and the necessity of adhering to them, since adhering to the Qur'an is not possible except through acting upon one's knowledge and the guidance of the Qur'an. The same is true about adhering to the Ahl al-Bayt (a).
In the Verse of Mubahala
The verse of Mubahala implies the superiority of Ahl al-Kasa' over other companions of the Prophet (s) as well; since, according to this verse, God has given the Prophet (s) mission to choose some people from among children, Muslim men and women to accompany him in Mubahala with the Christians of Najran; and the Prophet (s) chose Imam 'Ali (a), Lady Fatima (a) and Hasan (a) and Husayn (a).
There is no doubt that those are chosen for Mubahala who have great position in Faith and closeness to God; and in a Mubahala, one of the participants is the Prophet (s), those need to accompany him who are either similar in position with him or close to him than others regarding position. On the other hand, if there was some other Muslims who had such a high position in faith, the Prophet (s) would take him with himself as well, because there is not slightest possibility of injustice or unwise action about the actions of the Prophet (s).
Abu Riyah, the servant of Umm Salama narrated from the Prophet (s) that if there were other people more honorable than 'Ali (a), Fatima (s), Hasan (a) and Husayn (a), God would order me to do Mubahala with their help; and they are the best of people."
In Other Hadiths and Verses
Superiority of the Ahl al-Bayt (a) can also be understood from other hadiths and verses such as the verse of Mawaddah (42:23), hadith of Safina, hadith Bab al-Hitta, hadith Nujum, etc.