BEIRUT (AP) - Syria's opposition plunged into disarray Sunday as its president quit and its military chief refused to recognize the newly elected prime minister of an interim government for rebel-held areas. The moves reflected deep splits in the body the U.S. and its allies hope will emerge as the united face of the opposition and advance the fight to topple President Bashar Assad's regime. The missteps of the opposition's mostly exile political leadership drew little notice inside Syria, where rebel fighters dismissed it as ineffective and pushed ahead with their offensive to gain ground near the country's southern border with Jordan. Nearby, the Israeli military in the Golan Heights responded to fire by shooting back at targets inside Syria. The first blow to the opposition Syrian National Coalition was the surprise resignation of its president, who said he was quitting in frustration over what he called lack of international support and constraints imposed by the body itself. Mouaz al-Khatib, who rose to prominence as a preacher in Damascus' most famous mosque, said in a statement posted on his Facebook page that he was making good on an earlier vow to quit if undefined "red lines" were crossed. He also blamed world powers for not offering Syria's rebels the support they demand and complained that "international and regional parties" tried to push the Coalition toward negotiations with the Assad regime - something most members refuse. "All that has happened to the Syrian people - from destruction of infrastructure, to the arrest of tens of thousands, to the displacement of hundreds of thousands, to other tragedies - is not enough for an international decision to allow the Syrian people to defend themselves," the statement said. Despite electing a new, U.S.-educated prime minister last week to head a planned interim government, the Coalition has failed to make much of a mark inside Syria, where hundreds of independent rebel brigades are fighting a civil war against Assad's forces. Reflecting the growing dissension over that move, the head of the Coalition's military branch, Gen. Salim Idris, said his group refused to recognize the new prime minister, a little-known IT professional from Texas, because he lacked broad support among the opposition. "For the purpose of giving power to a prime minister to unite the revolutionary forces and lead the Syrian revolution toward certain victory, we unequivocally declare that the Free Syrian Army ... conditions its support and cooperation on the achievement of a political agreement on the name of a prime minister," Idris said in an online video. An aide to Idris, Louay Almokdad, said many prominent Syrian opposition figures opposed the election of Ghassan Hitto, who received 35 out of 48 votes cast by the Coalition's 63 active members. While al-Khatib's resignation surprised many Coalition members, some said it reflected problems that have caused five other members to resign in the past week. Coalition member Rima Fleihan told The Associated Press in Cairo that the body did not accurately represent Syrians. "We have problems internally with the structure of the Coalition and decisions being taken undemocratically," she said. Another recently resigned member, Walid al-Bunni, accused the Gulf state of Qatar, which heavily finances the opposition, of using pressure to install its candidate for prime minister. Others have accused the Muslim Brotherhood of exercising outsized influence. Late Sunday, the Coalition distributed a statement saying it had rejected the resignation and asked al-Khatib to keep doing his job. Secretary of State John Kerry said he regretted al-Khatib's resignation, but said it won't affect U.S. aid to the Coalition. Speaking to reporters during an unannounced trip to Baghdad, Kerry also said he had confronted Iraq, Syria's eastern neighbor, about allowing Iran access to its airspace for flights the U.S. believes are ferrying in weapons and fighters to the Assad regime. In a small victory for the opposition, senior Arab diplomats said they would transfer Syria's seat at the Arab League to the Coalition. The Syrian government's membership was suspended earlier in the crisis. The Coalition said it would send a delegation to a league summit that begins Tuesday in Qatar. The Syrian government, which contends the civil war is an international conspiracy being carried out by terrorists to weaken Syria, did not comment on the Coalition developments. Instead, it hosted a "National Dialogue Forum" in Damascus that included none of the forces seeking Assad's ouster. Few of the rebels inside Syria paid any attention to the exile opposition's problems, saying the Coalition had never done much for them anyway. "All this stuff that happens outside never makes any difference to us," rebel fighter Firas Filefleh said via Skype from the northern province of Idlib. He said he and his colleagues respect al-Khatib as a religious figure but that he and the Coalition were ineffective. "The Coalition has never made any difference for the fighting brigades," he said. They brought some flour and some canned goods but have never done more than that. Filefleh said he had no opinion of Hitto and said he had never heard of Gen. Idris, who purports to be the rebels' highest military leader. Late Sunday, the Coalition circulated videos it said showed Hitto during his first visit to Syria since his election. The videos showed Hitto in a sport coat and jeans, shaking hands in an unnamed town in Aleppo province. Meanwhile, rebels tried to advance their campaign to gain ground along the southern border with Jordan. Since last summer, the opposition has seized large swathes of land near the Turkish and Iraqi borders to the north and east, and has used them to organize and build supply lines. Victory in the south could allow them to do the same there. They have recently seized army checkpoints along a 15-mile (25-kilometer) strip of the border. The Britain-based Syrian Observatory for Human Rights said rebels clashed Sunday with forces at a checkpoint and military base in the area. Also Sunday, Israel's military said soldiers on patrol in the Golan Heights were fired upon and responded by firing back into Syria. It did not say if the Syrian fire was from rebels or the government. Rebels have been making gains on the Syrian side of the Golan Heights, a strategic plateau that Israel captured from Syria in the 1967 Mideast war and later annexed. The U.N. says more than 70,000 people have been killed since Syria's crisis began in March 2011. Associated Press reporters Aya Batrawy in Cairo, Matthew Lee in Baghdad and Albert Aji in Damascus, Syria, and Abdullah Rebhy in Doha, Qatar, contributed reporting.