UCH as Isaiah had dealt with the demise of the Assyrians in Isaiah 10, here, he speaks of the Babylonians in a similar way. Even before they were guilty of their evil deeds against God's people, they were destined for destruction. Again, although God was willing to use them for His purposes, they were still in much trouble for their own vices. It is apparent that the warnings and threats against Babylon had ramifications that reached far beyond the eventual empire of Nebuchadnezzar (Isaiah 13:10 & 11). The fall of Babylon during the days of Belshazzar was small potatoes up beside the fall of MYSTERY BABYLON THE GREAT in the last days of the kingdoms of this world, (that is, at the 2nd advent of Christ - Revelation 17 & 18). In that day the very Son of God, the King of the Jews, will lead the charge (Isaiah 13:4).
The day of the Lord is a designation which is frequently used in prophecy. There have been many days that (in a way) belonged to someone else, not the least of which was the day of Jesus' arrest (Luke 22:52-54). When Adam fell, when Cain killed Able, when Nimrod commenced building the tower of Babel (and on we could go); God was allowing the prerogative of the undeserving to prevail. But on the day of the Lord, only His wishes will matter. He will have His way in every way. When we pray, "Thy kingdom come, etc.," we are praying for the coming of the day of the Lord. Many will resist Him on that day. But, their rebellion will not succeed on any level (Isaiah 13:5-9). The voice that once shook Mt. Sinai will shake all of creation (Isaiah 13:13 & Hebrews 12:26). The annihilation of God's enemies will be intense and complete (Isaiah 13:16 & 18). The God who has restrained Himself from cleaning this planet with a second deluge of water (remember His rainbow promise?), will instead destroy His enemies with fire (Isaiah 13:19). Satan’s kingdom will fall!