Actual Crab Nebula distance
The star that went supernova and made the Crab Nebula is much farther
than Betelgeuse, about 6,500 light years away.
This paper estimates the distance at between 4,000 and 7,000 light
years, but in any event it was not that nearby.
http://www.hcs.harvard.edu/~jus/0302/bester.pdf
Nor was the brightest supernova ever recorded, in 1006 CE, at about
7,000 light years from Earth:
http://www.weblore.com/richard/1006_supernove.htm