Here is some evidence for that:
1- The Qur’an is the most superior of Allah’s holy scriptures (see 5:48), and the Prophet (s) informed this Ummah that the Ahulbayt (a) never separate from the Qur’an in Hadith al-Thiqlayn. This makes them higher than previous prophets because their knowledge and infallibility are intertwined with the Qur’an. They have the most knowledge after the Prophet (s), and that makes their rank higher.
2- 3:61 tells us Imam Ali (a) is the “self” of the Prophet (s), so we know he is higher than other prophets. Also remember that Imam Ali (a) has ilmul kitab (the full knowledge of the book).
3- Remember the hadith (accepted by Sunnis and Shia) that when the Mahdi (a) reappears and marches to Jerusalem, Prophet Isa (a) will pray behind him. This indicates Imam Mahdi (a) has a higher status than Prophet Isa (a), and we know that Prophet Isa is one of the 5 greatest messengers.
These are some brief proofs.