Our branch of Shia Islam, in that particular generation of the family, accepted the legitimacy of the eldest son, Isma‘il, as being the appointed Imam to succeed and that is why they are known as Ismailis.
(Forty-Ninth Hereditary Imam of the Shi‘i Isma‘ili Muslims)
Taken individually or collectively, these Seven Proofs demonstrate that the legitimate successor and true Imam after the Imam Ja‘far al-Sadiq is Mawlana Isma‘il ibn Ja‘far and that the Present and Living Imam of all Shi‘i Muslims must be the direct lineal descendant of Isma‘il ibn Ja‘far.
Proof #1: Imām Ja‘far designated Mawlānā Ismā‘īl as the next Imam by the rule of nass as per Twelver, Ismaili, Sunni and academic sources.
Proof #2: The only way to deny the nass of Mawlānā Isma‘il is through contradictoryhadiths presented in later Twelver hadith books.
Proof #3: Isma‘il’s death before Imam Ja‘far is not confirmed and may have been staged to protect him — as he was reportedly seen by eyewitnesses after his alleged death.
Proof #4: Even if Isma‘il had died before his father, the Imamat continued in Isma‘il’s son, Muhammad ibn Isma‘il, whom Isma‘il had appointed as his own successor.
Proof #5: Earliest Shi‘i hadiths lack the mention of Twelve Imams but instead predict exactly the first eighteen Imams in the Isma‘ili lineage of Imamat.
Proof #6: With the exception of the Nizari Ismaili Imamat, all other Shi‘i Imamat lineages have hidden Imams. This contradicts the Qur’anic definition of Imamat which requires the Imam always be present and manifest (mubin) in the world.
Proof #7: Imam Shah Karim al-Husayni Aga Khan IV is the forty-ninth hereditary Imam of Shi‘i Islam in direct, documented, lineal descent from Mawlana Isma‘il ibn Ja‘far al-Sadiq. As the only present (hadir), manifest (mubin) and living (mawjud) hereditary Imam, with a documented and validated lineage, Imam Shah Karim al-Husayni’s very existence is itself confirmation of his Imamat and that of his ancestors.
Click here to read at the source: http://ismailignosis.com/2014/10/02/who-succeeded-imam-jafar-al-sadiq-seven-proofs-for-the-imamat-of-imam-ismail-ibn-jafar/
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Ismailimail 
As an Ismaili, it is always good to hear or see the legitimacy of the line of Imamat; however faith itself should be enough for the believers. My concern of trying to “prove” this, may indicate/implicate a few things:
1) that we are not so sure ourselves and thus need to legitimize our following;
2) we need to prove to others – not sure why? – Inferiority complex?
3) we are putting down others who are born or have taken a different path – why? – Superiority Complex?
The Holy Qur’an has in many places indicated that “faith” is not something we simply get through whatever we believe, but it is a gift of Allah, given at Allah’s Choice and Mercy. Having said that, we; who have been guided to this and hopefully are gifted off the faith, should remain humble, thankful and pray that Allah’s Mercy should fall upon others too, while continuing to Guide & maintain us on sirat-ul-mustaqeem.
Please do not see this as derogatory of your efforts, but simply as a humble request and caution in how we approach things. We need to build on the pluralistic concept in Islam, in respecting others’ beliefs and allowing for inclusiveness without any elitism.
KH,
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Well said, Minaz. we don’t have to get into a discourse of proving. The world can see that by observing what Imam of the time is doing by His actions and guidance to His followers. We pray that others may also get to know the Imam of the time.
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I should have added – let such information belong to scholarly writings, where it belongs. Those who seek to know, can do so in their research. We, as Ismailies, can also read such scholastic papers and books, for our knowledge; but importantly let us all aspire to “know” the knowledge and turn it into “wisdom” in pursuing the Sufi path that our Tariqah is.
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