Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that:
$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A}\Pr[i]+ \sum_{j \in B}\Pr[j] +\sum_{i, j \in A}\Pr[i\ \textrm{AND}\ j] + \sum_{i, j \in B}\Pr[i\ \textrm{AND}\ j]$
My intuition for why this should be true is just by analogy to the simple inequality for $a, b \in \mathbb{R}:$ $2ab \leq a + b + a(a-1) + b(b-1)$ To see why this identity is true, note: $a + b + a(a-1) + b(b-1) = a^2 + b^2 = 2ab + (a-b)^2 \geq 2ab$
I don't have a proof for the more complex identity though. Is it true? Why does it hold?