König's theorem

''Therealsopropositiongraph theory called König's lemma.'\'

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In set theory, König's theorem states that if I issetm_in_icardinal numbersevery iI, and

then

The sum here isdisjoint union ofsets n_i; andproduct iscartesian product; we can similarly state itarbitrary sets (not necessarily cardinal numbers) by replacing < by strictly less thancardinality, i.e. therean injective function from m_in_i, but not one goingother way. The union involved need not be disjoint (a non-disjoint union can't be any bigger thandisjoint version, anyway).

(Of course thistrivial ifcardinal numbers m_in_ifinite andindex set Ifinite. If Iempty, thenleft sum isempty sumtherefore 0, whileright hand product isempty producttherefore 1).