Nowadays people make experiments and perform interbreeding to obtain new variation in animals. Like we have make horse and donkey interbreeding to produce a mule. Is this permissible, although Allah has sent all the living things in this world in pairs while people make from them new one?
Mules existed in the time of the Prophet (SAW) and he even owned many, and there doesn't seem to be any place where he discourages them or says that breeding them is forbidden:
The Prophet (ﷺ) did not leave anything behind him after his death except a white mule, his arms and a piece of land which he left to be given in charity. (Sahih Bukhari)
On the Day of Khaibar, the Messenger of Allah (ﷺ) prohibited eating domesticated donkeys, the meat of mules, every predator that possesses canine teeth, and every bird that possesses talons. (Jami at-Tirmidhi)
So, the default is that it is not forbidden.
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