(All emphasis is mine)
There is a difference between the "day of the Lord" and the "day of Christ".
In the Old Testament, the day of the Lord is God's wrath poured out on Israel. The NT expands it to the whole Earth.
In the New Testament, and specifically only Paul, reveals the phrase the "day of Christ". It is marked with a rejoicing for the saints. Wrath is not appointed.
Philippians 1:6 Being confident of this very thing, that he which hath begun a good work in you will perform it until the day of Jesus Christ:
Jesus delivered us from the wrath to come.
I Thes. 1:10 And to wait for his Son from heaven, whom he raised from the dead, even Jesus, which delivered us from the wrath to come.
We are present with Jesus Christ at His coming.
I Thes. 2:19 For what is our hope, or joy, or crown of rejoicing? Are not even ye in the presence of our Lord Jesus Christ at his coming?
Jesus Christ comes with all of His saints.
I Thes. 3:13 To the end he may stablish your hearts unblameable in holiness before God, even our Father, at the coming of our Lord Jesus Christ with all his saints.
Jude makes a similar pronouncement:
And Enoch also, the seventh from Adam, prophesied of these, saying, Behold, the Lord cometh with ten thousands of his saints,
We are alive in His coming. If we are alive at His coming, and we are with Him at His coming, then we are raptured before He comes. The sense here is that we are coming with Him, not to Him. He gathers before He comes. We who are alive are not preventing those who are asleep from coming with Him, but they rise first.
I Thes. 4:15 For this we say unto you by the word of the Lord, that we which are alive and remain unto the coming of the Lord shall not prevent them which are asleep.
16 For the Lord himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God: and the dead in Christ shall rise first:
Cross reference Corinthians for the Rapture:
I Cor. 15: 51 Behold, I shew you a mystery; We shall not all sleep, but we shall all be changed,
52 In a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed.
We are going to meet Christ in the air and the clouds, not on Earth.
17 Then we which are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord.
Paul does not use the day of Christ here, he uses the day of the Lord, along with wrath.
I Thes. 5:2 For yourselves know perfectly that the day of the Lord so cometh as a thief in the night.
Paul makes it clear we are exempt from the wrath.
I Thes. 5:9 For God hath not appointed us to wrath, but to obtain salvation by our Lord Jesus Christ,
Paul then hopes our triune humanity is preserved until the coming.
I Thes.5:23 And the very God of peace sanctify you wholly; and I pray God your whole spirit and soul and body be preserved blameless unto the coming of our Lord Jesus Christ.
Onto the next book! The day here in verse 10 is the day of the Lord, not the day of Christ.
II Thes. 1:10 When he shall come to be glorified in his saints, and to be admired in all them that believe (because our testimony among you was believed) in that day.
Look at the verses beforehand. Remember, the Lord comes with all of His saints.
I Thes. 1:6 Seeing it is a righteous thing with God to recompense tribulation to them that trouble you;
7 And to you who are troubled rest with us, when the Lord Jesus shall be revealed from heaven with his mighty angels,
8 In flaming fire taking vengeance on them that know not God, and that obey not the gospel of our Lord Jesus Christ:
9 Who shall be punished with everlasting destruction from the presence of the Lord, and from the glory of his power;
This is where it gets tricky.
Paul makes a distinction between His coming and His gathering of the saints. They are not the same thing.
II Thes. 2:2 Now we beseech you, brethren, by the coming of our Lord Jesus Christ, and by our gathering together unto him,
Paul says the day of Christ will come before the day of the Lord. Or else how can we comfort each other? In other words, he's telling Christians not to worry about the day of the Lord, as the day of Christ is coming before the day of the Lord.
2 That ye be not soon shaken in mind, or be troubled, neither by spirit, nor by word, nor by letter as from us, as that the day of Christ is at hand.
The day here in verse 3 is the day of the Lord, not the day of Christ, because the antecedent is not the verse beforehand, it is in the chapter before it. "As that" means "because".
3 Let no man deceive you by any means: for that day shall not come, except there come a falling away first, and that man of sin be revealed, the son of perdition;
The KJV is correct in saying the "day of Christ" rather than "day of the Lord". This is how the verse means in the KJV:
Do not be troubled by the day of the Lord (1:7-10), because ("as that") the day of Christ is coming.
Modern translations that change the day of Christ to day of the Lord because of their ignorance in grammar make the following translation mean:
Do not be troubled by the day of the Lord (1:7-10), because ("as that") the day of Lord is coming.