Once rescued the boys are expected to head straight to hospital in Chiang Rai town for further evaluations.
The rescue got off to a good start, with the first boy coming out at 5.40pm local time (11.40am United Kingdom time). It takes several hours.
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She believes the boys are still alive thanks to their coach. The operation is set to resume later.
But Thai authorities had already began there rescue efforts before the vessel was completed without Mr Musk's help.
Four of 12 schoolboys have been rescued so far from a flooded Thai cave after divers launched a daring and risky mission to free the children and their soccer coach, who were trapped underground for more than two weeks, Thai officials said.
Meanwhile rescuers fed a kilometres-long air pipe into the cave to restore oxygen levels in the chamber where the team was sheltering with medics and divers.
Foreign elite divers and Thai Navy SEALS were forced to begin the complex operation to extract the boys earlier than they had hoped on Sunday morning because forecast rainfall was threatening more flooding that might doom the mission. Officials did not immediately comment.
The divers formed part of the "daisy chain" of rescuers who led the four boys to the surface on Sunday.
He said the boys rescued Sunday are strong and safe but need to undergo detailed medical checks.
The dramatic and unsafe rescue at the Tham Luang cave in the northern province of Chiang Rai was launched on Sunday and four boys were brought out that day.
Authorities in northern Chiang Rai province began the unsafe mission to bring out the 12 boys and their football coach earlier on Sunday.
"The monsoonal pattern is present, meaning rain arriving in waves, which can't be adequately timed as they can escalate quickly", CNN meteorologist Gene Norman said Sunday evening.
Elon Musk, CEO SpaceX, have built a submarine to rescue the Thailand kids from the flooded Tham Luang cave.
On Sunday, Musk tweeted a series of videos showing a team of divers testing the pod in the swimming pool at Palisades Charter High School. If the tests come out successful, the sub would be placed on a 17-hour flight to Thailand.
Sam Teller, the spokesperson for the Boring Company, said four company engineers were "offering support in any way the government deems useful".
But efforts to get the others out were put on hold until Monday morning to give rescue workers time to replenish stocks of oxygen and other supplies.
The total number of boys pulled from the cave Monday is four, following the four that were pulled Sunday.
The area outside the Chiang Rai hospital was cordoned off by police.
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