John 15:4 - 11
ESV - 4 Abide in me, and I in you. As the branch cannot bear fruit by itself, unless it abides in the vine, neither can you, unless you abide in me. 5 I am the vine; you are the branches. Whoever abides in me and I in him, he it is that bears much fruit, for apart from me you can do nothing.
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According to the words of Jesus cited in the question, the sign of abiding in Christ is bearing fruit for Him, which to me is a consideration separate from what one believes or professes. In addition, Jesus said in Matthew 7:15-20 that even unbelievers would be known by their fruits, so the fruits that believers produce must also be of the right kind. The gospels contain multiple accounts of people who expressed belief in Jesus but were unwilling to either give up what they had to give up to be His disciple, or to endure the opposition and persecution that they would face as a result. (In the fruit analogy, they would be unable to endure the "pruning" by God of which Jesus spoke.) Therefore, despite their belief, they did not abide in Him and bear fruit for Him.
I think that the issue of abiding in Christ and being a believer are two separate issues. There is a good exposition of this passage in https://www.evidenceunseen.com/bible-difficulties-2/nt-difficulties/john-acts/jn-156-does-this-passage-teach-that-believers-will-be-sent-to-hell-for-not-bearing-fruit/, which in summary sets out three different branches in John 15. Two cases are believers; in one case they bear fruit, and in the other they do not. The third case is that of an unbeliever. Fruit is the result of believing, and there are many passages in the New Testament that reference God like a parent who disciplines His children whom He loves (Heb 12:4-11). If one is a believer and is not behaving the way God expects them to, they should expect discipline, whatever form that might take. That is what I think it looks like for a believer who is not abiding in Christ. Fruit are works, and works don't save you, faith does.
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