Not true for a large fraction of it. There is a reason it is called "reserve currency." It's used for trade.
Black markets the world over accept dollars.
Agreed the USD is the currency of choice for black markets across the world, but it too is subject to inflation. If the dollar loses value, purchase and exchange power is lost. Many smart investor watch the black market to judge true currency strength. So in the end, the idea that USD oversees mitigates inflation does not hold. I would direct you to this informative piece on inflation. The comments by Freepers are very good Link