‘Thus says the Lord Yahweh, “To the rejoicing of the whole earth, I will make you desolate. As you rejoiced over the inheritance of the house of Israel, because it was desolate, so I will do to you. You will be desolate, O Mount Seir, and all Edom, even all of it. And they will know that I am Yahweh.”
The whole earth is pictured as rejoicing over Edom's desolation, just as Edom had rejoiced over Israel's desolation. They had rejoiced over the fact that Israel had lost their inheritance, so they too will lose their inheritance at the hand of Yahweh (what of their gods then?). They will be a stark warning to all who afflict God's people.
‘You will be desolate, O Mount Seir, and all Edom, even all of it. And they will know that I am Yahweh.' The desolation will affect the whole of Edom and will be such that all who see it will recognise in it the hand of Yahweh. They will recognise His being and His power.
These predictions against Edom were literally fulfilled. Edom was first subjugated by Babylon, then by Medo-Persia, and then in 126 BC by John Hyrcanus, the Hasmonean ruler of Israel, who compelled them to become Jews and forcibly circumcised them (compare Obadiah 1:18). There is no trace of the Edomites in the modern day, although their desolate cities can still be identified (compare Jeremiah 49:13). Indeed the wonders of Petra have made it a tourist attraction.