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There are a couple different aspects to this. The first one the Bible gives us is found in Romans 10:1-4. “Brothers, my heart’s desire and prayer to God for them [the Jewish people] is that they may be saved. For I bear them witness that they have a zeal for God, but not according to knowledge. For, being ignorant of the righteousness of God, and seeking to establish their own, they did not submit to God’s righteousness.”
The Bible defines self-righteousness as relying on one’s own merits or righteousness as the grounds for their salvation and the reason for God to save them. The Bible teaches the exact opposite of this in many places, Ephesians 2:8 for example.
It also defines the behaviors often exhibited by individuals that are self-righteous. The Bible gives an example of this in Luke 18:9-14. The Pharisee felt that he was better than others, and lifted himself up by pointing out all of the issues in other people’s lives. Self righteousness leads one to feel superior than those around him. This feeling of superiority often produces acts of rudeness and pride that hurt others.
We need to be sure that we are trusting fully in the merits of Christ’s perfect life and depending completely on His sacrifice at Calvary, otherwise we too may become self-righteous like the Pharisee. The only way we can truly know ourselves is if we look constantly to Christ and behold His character. We need to see His perfection, and our own imperfection in contrast, in order to be safe.