Why is a Greek default negative for EUR?

The threat of a Greek default is sending the Euro lower. Why is that?

I get where a Greek default would absolutely send the price of Greek bonds lower (since Greece is unlikely to pay), and if the ECB promised unlimited bailouts of Greece, the Euro would also go lower (since ECB would just print all these Euros to bail out Greece), but if Greece actually defaulted, why is that damaging to the Euro?

It seems that, if you have a Greek bond, it is because you want more Euros. You bought the bond, denominated in Euros, in order to collect the interest payments, also denominated in Euros. You traded Euros for the bond, and the bond promises more Euros. If Greece defaults on the bond, I see where that makes the bond worth less (or worthless), but I don't follow how that makes the Euro drop.