Corollary 1

A graph $\Gamma = (A, S)$ is connected if and only if there does not exist a partition $(S', S'')$ of $S$ into two non-empty subsets such that no element of $S'$ is joined in $\Gamma$ to any element of $S''$.

Suppose that $\Gamma$ is not connected and let $S'$ be one of its connected components. The set $S'' = S - S'$ is non-empty by Prop. 1, (i) and no element of $S'$ is joined to any element of $S''$ by Prop. 1, (iii).

Suppose now that $\Gamma$ is connected and let $(S', S'')$ be a partition with the stated property. By Prop. 1, (iv), the set $S'$ contains at least one connected component, so $S' = S$ and $S'' = \varnothing$, a contradiction.