No, that’s not so at all.
Pride is the source of all sin, and pride and selfishness go hand-in-hand.
You will lose this one big time. Go back even to the beginning. The real reason for the fall of man was not only because of disobedience, but the selfish and self-centered desire to “be like God”. Remember the admonition of Paul in Romans 12:1- “ Therefore, I urge you, brothers, in view of God’s mercy, to offer your bodies as living sacrifices, holy and pleasing to God - this is your spiritual act of worship.” And of course, “God’s act of mercy” was the ultimate act of self-denial when Christ, blamesless in all respects, offered himself up as an atoning sacrifice for our sins on the cross. In fact the whole Bible is about the self-centeredness of man being the primary sin separating us from God, and his call for our self-denial to acknowledge him as God and denial of ourselves in the treatment of other people.