Shias believe the successor of Muhammad has to be from the Ahl-ul-Bayt( Family of the Prophet). Abu Bakr as-saddiq and Umar ibn al-Khattab were father in laws of the Prophet. Othman ibn Affan is also related and Ali was the Prophet's cousin and son in law. Can this be used to argue with their opinion that the successors of Muhammad have to be from Ahl-ul-Bayt?
Only a specific group of relatives of the Prophet are considered to be Ahl-ul-Bayt. When Shias refer to Ahl-ul-Bayt they do not mean all relatives of the Prophet but this specific group.
As authentic Hadith confirm, Ahl-ul-Bayt are the direct family members of the Prophet: Ali ibn Abi Talib, Fatimah bint Muhammad and their two sons Hassan and Hussain ibn Ali. They are considered to be mahsum, without sin.
Sahih Muslim Book 031, Hadith Number 5955. Chapter : The merits of the family of the Prophet (may peace be upon him).
'Aisha reported that Allah's Apostle (may peace be upon him) went out one morning wearing a striped cloak of the black camel's hair that there came Hasan b. 'Ali. He wrapped him under it, then came Husain and he wrapped him under it along with the other one (Hasan). Then came Fatima and he took her under it, then came 'Ali and he also took him under it and then said: Allah only desires to take away any un-cleanliness from you, O people of the household, and purify you (thorough purifying)
The other nine divinely chosen successors of the Prophet are descendants of Hussain ibn Ali and are also included in the household of the Prophet.
Not really, since the primary reason Shi'ites consider Ali the rightful successor to the prophet rather than Abu Bakr, Umar and/or Uthman isn't because of (or at least not exclusively because of) the Ahl-ul-Bayt argument you're trying to defuse, rather it's because of the hadith of Ghadir Khumm:
Of whomsoever I had been Master, Ali here is to be his Master.
Shi'ites take this to mean that Ali was explicitly and personally appointed to succeed the prophet — by the prophet himself — while Abu Bakr, Umar and Uthman were not.
بسم الله الرحمن الرحیم
As @Noor mentioned well, only a specific group of Prophet Muhammad's relatives are deemed to be Ahlul-Bayt, all of them. In truth, Ahlul-Bayt of Prophet Muhammad (SAWW) only consist of the following individuals:
• Fatimah al-Zahra (s.a.),
• Imam Ali (a,s,),
• Imam Hasan (a.s.),
• Imam Husayn (a.s.),
• 9 descendants of Imam Husayn (a.s.).
And of course including the Prophet (SAWW) himself, they will become fourteen individuals. Of course, at Prophet Muhammad’s (pbuh) time only 5 of them (including Prophet Muhammad) were living and actually the rest weren’t born yet.
Source and further information: