Union is Dominated by Disjoint Union

Theorem

Let $I$ be an indexing set.

For all $i \in I$, let $S_i$ be a set.

Then:

$\ds \bigcup_{i \mathop \in I} S_i \preccurlyeq \bigsqcup_{i \mathop \in I} S_i$

where $\preccurlyeq$ denotes domination, $\bigcup$ denotes union, and $\bigsqcup$ denotes disjoint union.

Proof

For all $\ds x \in \bigcup_{i \mathop \in I} S_i$, there exists a $\map i x \in I$ such that $x \in S_{\map i x}$.

Thus the mapping $\ds \iota : \bigcup_{i \mathop \in I} S_i \to \bigsqcup_{i \mathop \in I} S_i$ defined by:

$\map \iota x = \tuple {x, \map i x}$

is an injection.

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$\blacksquare$