Kőnig's theorem (set theory)
= Kőnig's theorem (set theory)
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= König theorem for cardinal numbers
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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.