The Welfare Theorems

Choose an arbitrary alternative $B$, and consider a profile in which $B$ never takes an intermediate position in any individual's ranking; it is either first or last. Then society must rank $B$ either first or last. Suppose instead that society ranks $A\succeq B\succeq C$. Since $B$ is either at the top or the bottom for any individual, $C$ can be moved above $A$ without disturbing her $B,C$ or her $A,B$ ranking. So IIA implies $A\succeq B$ and $B\succeq C$ in the new social order. Transitivity implies $A\succeq C$ in the new order. Finally, unanimity implies $C\succ A$ in the new order, a contradiction. So $B$ must be extremal.

Some voter $n^*=n(B)$ is pivotal in that by changing his vote at some profile he can move $B$ from the bottom to the top. To see this, start with a profile that puts $B$ at the bottom for every voter. Then by unanimity, $B$ must be at the bottom of the social ranking. Now move $B$ in each voter's profile, one by one, from bottom to top. Let $n^*$ be the first voter whose change causes $B$ to move off the bottom. Since $B$ is extremal in every ranking, it must now be at the top of the social order. There must be one such $n^*$ because if all voters have switched, unanimity requires that $B$ be at the top. Denote by $\succ'$ the profile wherein $n^*$ puts $B$ at the bottom, and $\succ''$ the profile after $n^*$ switches.