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Usyk dethroned Joshua of the WBA (super), IBF and WBO world heavyweight titles in Tottenham last weekend.
He put on a masterclass to beat the Brit, winning a unanimous decision.
A rematch between Usyk and Joshua is reportedly in the works, with Fury offering his heavyweight rival a piece of advice ahead of their second meeting.
On Joshua vs. Usyk, the WBC title holder said: "Did I watch the fight? Yes I did.
"Was I absolutely wounded that he won? Yes I was.
"I was hoping Joshua could win the fight, but he couldn't - and that's none of my business."
Fury added: "The only thing I'm bothered about is beating Deontay Wilder, and that's the most dangerous heavyweight out there.
"In my opinion, Wilder beats Joshua, Usyk, all the rest of the division, comfortable - but he cannot beat me.
"Usyk did his job, he had to do what he had to do, and that's that, and Joshua has got to do what he has got to do.
"After him, we will talk, the promoters will do their job, and I will always do mine.
"Never worry about the 'Gypsy King' fulfilling his end of a bargain - I will always fucking fight until there's not a fight left in me. You just worry about the other people doing their end of the bargain."
Asked what advice he would give to 'AJ' ahead of facing Usyk again, Fury replied: "My advice to Joshua in the rematch is get stuck in the best way you know how, put his best foot forward and swing away."
Joshua's crushing defeat to Usyk delays a potential undisputed clash with Fury, who returns to action next weekend.
'The Gypsy King' faces Deontay Wilder in a trilogy bout in Las Vegas.
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