"All the thirteen Wild Boars are now out of the cave", the SEAL unit said on its Facebook page, adding all were safe.
The four boys who were recovered on Sunday are now eating normal food, Jedsada said.
So far eight boys aged between 12 and 16 have been led out of the cave system by a team of expert divers over two days.
A crack team of foreign divers and Thai Navy SEALS guided the boys during a nine-hour operation through almost 4 km (2.5 miles) of sometimes submerged, pitch-dark channels from where they have been trapped for more than a fortnight.
"All five will be brought out at the same time today", he said, to cheers from reporters and rescue.
The final part of the operation to rescue a boys' football team from a cave in Thailand has begun.
The footballers still awaiting rescue are about two kilometers further in, at a point very hard to access.
But crews continue to work with the knowledge that additional monsoon rains could threaten the operation and put the remaining kids and their coach in peril. But officials waited several hours before confirming their rescue. Some of the parents told Reuters they had not been told who had been rescued and that they were not allowed to visit the hospital. The entire Wild Boar youth soccer team and its coach have been safely removed from a cave in Thailand.
Chiang Rai Gov. Narongsak Osatanakorn says the rescue mission began at 10.08am and involves 19 divers. "The big heroes in this are the children and the four Thai navy Seals who were looking after them".
They had been trapped inside the Tham Luang cave for more than two weeks.
Four more of the boys were carried on stretchers out of the labyrinthine Tham Luang cave on the Myanmar border on Monday, bringing to eight the total number brought out so far after two rescue pushes in successive days.
Four boys and their 25-year-old football coach "will be extracted on Tuesday". What happens when the players are out of the cave " The kids are footballers so they have high immune systems", Jesada said.
The second group of 4 that were pulled out afterwards are aged between 12 and 14.
Four boys and their soccer coach remain in the cave.
The soccer team first made contact with rescuers from the United Kingdom last Monday. The heaviest rain tends to fall in late summer.
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.
They have also been given sunglasses to wear as they adjust to daylight after days trapped in the dark.
The rescue mission is still underway.
In that mission, two divers accompanied each boy, guided by a rope.
"Just returned from Cave 3", tweeted the CEO. He posted photos of the cave interior and a video showing people working their way through chest-high water.
SpaceX and Tesla chief executive Elon Musk said he went to the cave with a miniature submarine to try and help the boys.
Divers have rescued eight of the boys.
"They're asking for chocolate".
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