A Sahâbî (Companion) is anyone that met the Prophet (sallAllâhu ‘alayhi wa sallam) while believing in him, and dying upon that condition.
The Companions (Sahâbah) of the Prophet (sallAllâhu ‘alayhi wa sallam) are the best from all those that accompanied the prophets, due to the statement of Allâh’s Messenger (sallAllâhu ‘alayhi wa sallam): “The best of people are my Companions.”  Al-Bukhârî and others have reported this hadîth.
The best among the Sahâbah are the Muhâjirûn (those who performed the migration from Makkah to Madînah), due to their combining between migrating and giving support (in Allâh’s Cause). Then after them, are the Ansâr.
The best among the Muhâjirûn are the four righteous Khalîfahs: Abu Bakr, ‘Umar, ‘Uthmân and ‘Alî.
Abu Bakr – He is “As-Siddîq” ‘Abdullâh bin ‘Uthmân bin ‘Âmir from the Tribe of Taym bin Marrah bin Ka’ab. He was the first male to believe in the Messenger of Allâh (sallAllâhu ‘alayhi wa sallam). He was his companion in the migration (Hijrah) and his deputy during times of prayer and Hajj. He was his Khalîfah (designated successor) for his nation. Five of those who were given the glad tidings of Paradise accepted Islâm through his hands.
They are ‘Uthmân, Az-Zubair, Talha, ‘Abd-ur-Rahmân bin ‘Awf and Sa’ad bin Abî Waqqâs. He died in Jumâdal-Al-Akhirah, in the 13th year of Hijrah, when he was 63 years old. The above five along with Abu Bakr, ‘Alî bin Abî Tâlib and Zayd bin Hârithah form the eight that were the first of people to accept Islâm. Ibn Ishâq stated this and what he meant by it, was that they were the first from amongst the males, after (the advent of) the Message.
‘Umar – He is Abu Hafs “Al-Fârûq” ‘Umar Ibn Al-Khattâb from the tribe of ‘Adiyy bin Ka’ab bin Lu’â. He accepted Islâm in the sixth year after the Prophet (sallAllâhu ‘alayhi wa sallam) was sent, after nearly forty men and eleven women (had accepted Islâm). The Muslims were overjoyed by that occasion and Islâm spread in Makkah afterward. Abu Bakr entrusted him with the leadership of the Muslim nation, and so he rose to the hardship of the Khilâfah in the best manner. This was until he was killed and died as a martyr in Dhul-Hijjah of the year
23 H, at the age of 63.
‘Uthmân – He is Abu ‘Abdillâh “Dhun-Nûrain” ‘Uthmân bin ‘Affân from the tribe of Umayyah bin ‘Abd-ish-Shams bin ‘Abd Manâf. He accepted Islâm after the Prophet’s entrance into Dâr-ul-Arqam. He was a rich and generous man. He was given the Khilâfah after ‘Umar Ibn Al-Khattâb (radyAllâhu ‘anhu), due to the agreement of the members of the governing body. He remained Khalîfah until he was killed and martyred in Dhul-Hijjah in the year 35 H. He was 90 years old at the time of his death, according to one of several opinions.
‘Alî – He is Abul-Hasan ‘Alî bin Abî Tâlib. The name of Abu Tâlib was ‘Abd Manâf bin ‘Abd-il-Muttallib. He was the first from the youth to accept Islâm. The Messenger of Allâh (sallAllâhu ‘alayhi wa sallam) gave him the banner on the Day of Khaibar and Allâh granted victory at his hands. He was given the Khilâfah after the murder of ‘Uthmân. Thus, he became the Khalîfah, governing until he was killed himself, and martyred in the month of Ramadân in the 40 H, at the age of 63 .
The best of these four is Abu Bakr, then‘Umar, then‘Uthmân, then‘Alî. This is based on the report of Ibn ‘Umar (radyAllâhu ‘anhu):
“We used to make preference between the people during the time of the Prophet (sallAllâhu ‘alayhi wa sallam). So we preferred Abu Bakr, then ‘Umar Ibn Al-Khattâb, then ‘Uthmân bin ‘Affân.” This was reported by Al-Bukhârî. 
It is also based on (the hadîth) in Abu Dawûd:
“We used to say, while the Messenger of Allâh (sallAllâhu ‘alayhi wa sallam) was alive: The best of the Prophet’s ummah after him, is Abu Bakr, then ‘Umar, then ‘Uthmân.” At-Tabarânî added to this in his report: “The Prophet would hear that and he would not oppose it.”
I did not find the wording that the author mentions, in which there is the addition of ‘Alî bin Abî Tâlib. 
The most deserving from them of the Khilâfah, after the Prophet, was Abu Bakr (radyAllâhu ‘anhu). This is since he was the best of them and foremost to them in accepting Islâm. Also, the Prophet (sallAllâhu ‘alayhi wa sallam) preferred him to lead the prayer. Furthermore, the Sahâbah agreed upon his superiority and upon giving him the oath of allegiance (bay’ah). And it is not befitting for Allâh to unite them in agreement upon an error. Then ‘Umar, since he was the best of the Sahâbah after Abu Bakr and because Abu Bakr entrusted the Khilâfah to him. Then ‘Uthmân, due to his merits and due to his election by the Majlis Ash-Shûrâ (governing body). They are the ones mentioned in this verse from a poem:
“‘Alî, ‘Uthmân, Sa’ad and Talha. Zubair and Dhu ‘Awf — members of the governing body.”
Then comes ‘Alî, due to his merits and the consensus of the Muslims of his time upon it. These four are the rightly guided Khalîfahs, whom the Prophet (sallAllâhu ‘alayhi wa sallam) said concerning them:
“Stick to my Sunnah and the Sunnah of the rightly-guided Khalîfahs after me. Bite onto it (the Sunnah) with your molar teeth.”
And he (sallAllâhu ‘alayhi wa sallam) said: “The Khilâfah after me will endure for thirty years.” Ahmad, Abu Dawûd and At-Tirmidhî reported this hadîth and Al-Albânî said that its chain of narration was hasan. Thus, the last part of it, was the Khilâfah of ‘Alî. This is as the author has stated and it is as if he places the Khilâfah of Al-Hasan subordinate to his father’s or just doesn’t acknowledge it (i.e. Al-Hasan’s Khilâfah) at all, due to his yielding it.
The Khilâfah of Abu Bakr (radyAllâhu ‘anhu) endured for two years, three months and nine nights, from the 13th of Rabî’-ul-Awwal 11 H to the 22nd of Jumâdal-Akhirah 13 H.
The Khilâfah of ‘Umar (radyAllâhu ‘anhu) lasted ten years, six months and three days, from the 23 rd of Jumâdal-Akhirah 13 H to the 26th of Dhul-Hijjah 23 H.
The Khilâfah of ‘Uthmân (radyAllâhu ‘anhu) lasted twelve years minus 12 days, from the month of Muharram 24 H to the 18th of Dhul-Hijjah 35 H.
The Khilâfah of ‘Alî (radyAllâhu ‘anhu) lasted four years and nine months from the 19th of Dhul-Hijjah 35 H to the 19th of Ramadân 40 H.
Combined, the Khilâfah of these four (Companions) lasted twenty-nine years, six months and four days.
Thereafter, Al-Hasan bin ‘Alî (radyAllâhu ‘anhu) was given the oath of allegiance on the day that his father, ‘Alî died. Then, in the month of Rabî’-ul-Awwal 41 H, the command was given to Mu’awiyah (radyAllâhu ‘anhu) and due to that, the signs which the Prophet (sallAllâhu ‘alayhi wa sallam) spoke of in his statement: “The Khilâfah after me will endure for thirty years”and his statement: “Indeed, this son of mine is a Sayyid (noble) and perhaps it will be that Allâh will bring peace between two large parties (i.e. that of Al-Hasan and Mu’awiyah) through him”came to pass.
Al-Hasan was the grandson of the Messenger of Allâh (sallAllâhu ‘alayhi wa sallam) and the fruit of his offspring. He was the Amîr-ul-Mu’minîn (Commander of the Believers), son of the Amîr-ul-Mu’minîn ‘Alî bin Abî Tâlib. He was born on the 15 th of Ramadân in the third year after Hijrah. He died while in Madînah and was buried in the graveyard Al-Baqî’ in Rabî-ul-Awwal 50 H.
Al-Husayn was the grandson of the Messenger of Allâh (sallAllâhu ‘alayhi wa sallam) and the fruit of his offspring. He was the son of ‘Alî bin Abî Tâlib. He was born in Sha’bân in the fourth year after Hijrah and was killed at Karbala on the tenth of Muharram 61 H.
 Sahîh Al-Bukhârî: Book of the Virtues of the Companions of the Prophet (no.3651 ) and Sahîh Muslim: Book of the Virtues of the Companions (no. 2533) from the hadîth of Ibn Mas’ûd (radyAllâhu ‘anhu). It is also reported on the authority of ‘Imrân bin Husayn (radyAllâhu ‘anhu) in Al-Bukhârî (no.2561 ,3650 , 6438 &6695 ) and Muslim (no.2535 ). Abu Hurairah (radyAllâhu ‘anhu) also reported it in Muslim (no.534 ) as well as others. Furthermore, this is a mutawâtir hadîth. Al-Hâfidh Ibn Hajr stated the correctness of it being mutawâtir in the introduction to his Al-Isâbah.
 Sahîh Al-Bukhârî: Book of the Virtues of the Companions (no.3655 ). And in another wording found in Al-Bukhârî (no.3697 ), it states:
“During the lifetime of the Prophet, we used to not put anyone equal to Abu Bakr, then (the same was with) ‘Umar, then (the same was with) ‘Uthmân. Then we left the (remaining) Companions of the Prophet alone, not distinguishing between any of them.”
 Isnâduhu Sahîh: This is an authentic hadîth reported by Abu Dawûd (4628), At-Tirmidhî (3707) and Ibn Abî ‘Âsim in As-Sunnah (1190). Its chain of narration (isnâd) is authentic as has been stated by Al-Albânî in his checking to As-Sunnah (2/567). As for the addition that Shaikh Ibn Al-‘Uthaimîn has mentioned, which is found in At-Tabarânî, which is: “ The Prophet would hear that and he would not oppose it”, then it is an authentic addition, which is established in many paths of narration found in Ibn Abî ‘Âsim (1194,1195 , 1196 &1197 ), Ahmad (2/14) and others with authentic chains. Refer to the checking of Ibn Abî ‘Âsim’s As-Sunnah (2/568-569) as well as Fath-ul-Bârî (7/16-17)
 Sahîh Al-Bukhârî: Book of Peace Making (no.2704 ) from the hadîth of Abu Bakrah (radyAllâhu ‘anhu).
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