That is easily shown false. Cornelius is a great example that contradicts this.
Acts 10:1 There was a certain man in Caesarea called Cornelius, a centurion of the band called the Italian band,
2 A devout man, and one that feared God with all his house, which gave much alms to the people, and prayed to God alway.
3 He saw in a vision evidently about the ninth hour of the day an angel of God coming in to him, and saying unto him, Cornelius.
4 And when he looked on him, he was afraid, and said, What is it, Lord? And he said unto him, Thy prayers and thine alms are come up for a memorial before God.
5 And now send men to Joppa, and call for one Simon, whose surname is Peter:
6 He lodgeth with one Simon a tanner, whose house is by the sea side: he shall tell thee what thou oughtest to do.
Cornelius wasn't saved here. Yet he was very devout and feared God. He gave alms and prayed to God always. But he was not saved and did not receive the Holy Spirit until Peter came and preached to him.
Acts 10:30 And Cornelius said, Four days ago I was fasting until this hour; and at the ninth hour I prayed in my house, and, behold, a man stood before me in bright clothing,
31 And said, Cornelius, thy prayer is heard, and thine alms are had in remembrance in the sight of God.
32 Send therefore to Joppa, and call hither Simon, whose surname is Peter; he is lodged in the house of one Simon a tanner by the sea side: who, when he cometh, shall speak unto thee.
33 Immediately therefore I sent to thee; and thou hast well done that thou art come. Now therefore are we all here present before God, to hear all things that are commanded thee of God.
34 Then Peter opened his mouth, and said, Of a truth I perceive that God is no respecter of persons:
35 But in every nation he that feareth him, and worketh righteousness, is accepted with him.
36 The word which God sent unto the children of Israel, preaching peace by Jesus Christ: (he is Lord of all: )
37 That word, I say, ye know, which was published throughout all Judaea, and began from Galilee, after the baptism which John preached;
38 How God anointed Jesus of Nazareth with the Holy Ghost and with power: who went about doing good, and healing all that were oppressed of the devil; for God was with him.
39 And we are witnesses of all things which he did both in the land of the Jews, and in Jerusalem; whom they slew and hanged on a tree:
40 Him God raised up the third day, and shewed him openly;
41 Not to all the people, but unto witnesses chosen before of God, even to us, who did eat and drink with him after he rose from the dead.
42 And he commanded us to preach unto the people, and to testify that it is he which was ordained of God to be the Judge of quick and dead.
43 To him give all the prophets witness, that through his name whosoever believeth in him shall receive remission of sins.
44 While Peter yet spake these words, the Holy Ghost fell on all them which heard the word.
45 And they of the circumcision which believed were astonished, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Ghost.
46 For they heard them speak with tongues, and magnify God. Then answered Peter,
47 Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we?
48 And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days.
Cornelius wasn't saved, yet he was very devout and sought God. And Peter in verse 35 says that all men everywhere who fear God and work righteousness are accepted with God.
So there you go right there, this shows the unsaved man is not a slave to sin, and can indeed do good works. They are not all in rebellion to God.
It was only after Peter preached to Cornelius of Christ that the Holy Spirit fell upon him in verse 44. Notice it says "which heard the word". So, as always in scripture, this shows you first hear the word of God, then believe, then receive the Holy Spirit (vs 47). And what exactly did Cornelius hear? Read verse 43, that if he were to believe on Christ he would receive remission or forgiveness of sins.
There is not one word of Cornelius being regenerated to believe here. He was a very devout man who feared God. He did not receive the Holy Spirit until he first heard and believed the gospel from Peter.
The doctrine of Total Depravity as held by Calvinists is error and easily shown so by much scripture.