König theorem for cardinal numbers
= König theorem for cardinal numbers
{c}
{wiki=König's_theorem_(set_theory)}
If $\kappa_i<\lambda_i$ for every $i\in I$, then
$$
\sum_{i\in I}\kappa_i<\prod_{i\in I}\lambda_i.
$$
The injection uses functions supported at one coordinate. For non-surjectivity, given any map from the disjoint union to the product, choose at coordinate $i$ a value omitted by the $i$th row; the resulting diagonal element is outside its image.