We may be one of the 'major world powers' (after a handful of others) but that is partly due to our EU membership in itself.
It also begs the question as to why we joined in the first place. As a country we were struggling badly and figured out that it would be better for us to join in a large trading bloc. The problem was that after WWII we thought we could simply sit back and reap the victor's rewards and carry on trading to the diminishing Empire/Commonwealth as we had previously done. We were a great trading nation, and always would be. Except, we weren't: Sitting back on their laurels, the bosses of British companies were endlessly complacent and failed to invest and fell far behind other countries in terms of competitiveness.
What has changed since then? I still don't see British companies investing massively in order to gain competitive advantage. But that's ok, because we are a world power. Surprisingly, that isn't enough...