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Naming & Laqabs of Imam Zain ul Abideen; Why called Zain ul Abideen

The First son of Imam Hussain (A.S) 

Imam Zain ul Abideen (A.S) was the first son who was born to Imam Hussain (A.S) on the 15th of Jamadi ul Awwal in 38 Al Hijri. He was named as Ali.

And so the process began to keep this name alive in the house of Imam Hussain (A.S) . So every child which was born was named as Ali like Ali Akbar (A.S) and Ali Asghar (A.S) . Because the enemies of Islam and those who were envious to the house of Muhammad (PBUH) had always tried to remove this name Ali
from the world and the events which happened afterwards proved this. But they didn't know how could they remove the name of a person from the earth who had been with the Prophet Muhammad (PBUH) all his life, who stayed with him when every one was against him and whose son was going to
be the Hujjat Al Qaim (A.S) on earth.

Laqabs of Ali Zain ul Abideen (A.S) 

His Laqab was Zain ul Abideen, Syed us Sajideen, Sajjad and Zo al Safnat. His kuniat was Abu Muhammad.
The real reason for his being Zain ul Abideen is that Prophet Muhammad (PBUH) himself called him by this laqab and had said in his life that when Zain ul Abideen (A.S) will be called at the Day of Judgement, my son Ali ibn Hussain (A.S) will say Labaik and will bow before Allah.