 Solution

König theorem for cardinal numbers says that for families of cardinals ${\kappa }_i<{\lambda }_i$ indexed by a set $I$,The sum is the cardinality of the disjoint union of sets $A_i$ of sizes ${\kappa }_i$, and the product is the cardinality of ${\prod }_i{B}_i$ for sets of sizes ${\lambda }_i$.

Consider any function $F:{⨆}_i{A}_i\to {\prod }_i{B}_i$. For each $i$, the set of $i$th coordinates of images of points from the $i$th block has cardinality at most ${\kappa }_i<{\lambda }_i$. Choose $b_i\in B_i$ outside it. Then $b=(b_i{)}_i$ is not the image of any point: a point from block $i$ has the wrong $i$th coordinate. Thus there is no surjection from the sum set onto the product set. Cardinal comparability in $ZFC$ makes the inequality strict in the indicated direction. For the empty index set the statement is $0<1$, also true. This is the diagonal argument underlying the inequality.