Nope. The O-Ring leaves play and your life goes to 20 both while Lich's Mirror's ability is resolving. The O-Ring's trigger goes on the stack, then resolves after the Mirror's ability is finished resolving. Thus, Sanguine Bond won't be in play when you gain life.
Well, the oracle rulings are down right now, but I'm pretty sure setting a life total to a specific number, like with Magister Sphinx, is just a short-cut, and actually means gaining or losing the appropriate amount of like.
A buddy and I were thinking about this, but we couldn't figure it out. At first, we thought if you lost by having 0 life with both out, you would gain 20 life and cause 20 life loss, but then, as we re-read Lich's Mirror, it would seem that Sanguine Bond would be shuffled in before the life gain, so the effect would never trigger. Our thought is that there is no way to get this to work, since Sanguine Bond will ALWAYS be shuffled into the library before the life gain would stack, but any clarification/explanation will be appreciated.
Oh well...:rollout:
Hey, there's an idea! Does this achieve the desired result?
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Ah, well. Thanks.