FLP86's famous proof regarding impossibility of async distributed processes with a single fault assumes in the proof of the third lemma the existence of an event $e'$ such that the neighbor configurations $C_0$ and $C_1$ can be related as
$C_1 = e'(C_0)$.
I don't get how this is possible, as it seems to me like $e'$ carries out a state transition from a 0-valent configuration to a 1-valent configuration. In addition to this being counter intuitive the proof of case 1 of lemma 3 clearly states that any successor of a 0-valent configuration has to be a 0-valent configuration. What am I missing here?