The dispute over ElBaradei underlines the fragmentation of Egypt's politics as the country continues to be roiled by bout after bout of unrest and violence since Mubarak's ouster.
The 2011 uprising opened the way for the political rise of the Muslim Brotherhood, which was long suppressed by Mubarak's Western-backed regime, and Morsi was elected last year by a narrow margin. The fundamentalist movement swiftly rejected ElBaradei's appointment.
The Brotherhood has vowed to boycott the political process, saying the military maneuver was a coup that overturned a democratically elected government.
"Now it's clear that the Mubarak regime has the upper hand," Brotherhood spokesman Ahmed Aref alleged. "We cannot accept the strategy of arm twisting; we cannot accept the authority being snatched by force," he told The AP.
The group's powerful deputy Khairat el-Shater, former leader Mahdi Akef, Rashad Bayoumi and Saad el-Ketatni have been accused of inciting violence against protesters in Cairo.
The silver-haired new president, meanwhile, insisted national reconciliation was his top priority.
"Enough already with divisions," he told reporters on Saturday. "We need to mobilize our forces to build this nation," he said. He also called on the Brotherhood to join the political process. "The Brotherhood is a part of this nation, if they decide to join, we will welcome them."
"I want everyone to pray for me. Your prayers are what I need from you," he told worshippers on Friday in comments published Saturday by the independent el-Tahrir daily.
On Saturday, he met with el-Sissi and Interior Minister Mohammed Ibrahim, who is in charge of the police. Later he met with the three young leaders of Tamarod, or Rebel, which organized the massive opposition protests that began June 30.
Despite his words, both sides braced for the possibility of more violence as Egypt's political unraveling increasingly left little room for middle ground or dialogue.
In the northern part of the Sinai Peninsula, gunmen shot dead a Christian priest while he shopped for food in an outdoor market on Saturday. It was not immediately clear if the shooting was linked to the political crisis, but minority Christians have faced increased attacks in the wake of the Islamist rise to power in the nation of 90 million people.
South of the Sinai city of el-Arish, security officials said suspected Islamic militants bombed a natural gas pipeline to Jordan. The attacks early Sunday on two points on the pipeline started fires that were soon put out, but the flow of gas was disrupted, said the officials, who spoke on condition of anonymity because they were not authorized to speak to the media.
It was the first attack on Egypt's natural gas pipelines in Sinai in over a year.
In Cairo's eastern suburb of Nasr City near the Rabaah al-Adawaiya mosque — the main rallying Muslim Brotherhood rallying point — lines of fighters brandished homemade weapons and body armor at road blocks decorated with Morsi's picture.
"The people here and in all of Egypt's squares are ready for martyrdom to restore legitimacy," said Abdullah Shehatah, a senior leader of the Freedom and Justice Party, the Brotherhood's political arm. "This coup and all its institutions are illegal."
Next door in the relatively upscale Heliopolis district, people chanted against Morsi and honked car horns in appreciation of roadblocks manned by Egypt's military.
Security forces boosted their presence with armored personnel carriers and checkpoints across the nation's capital.
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