Union is Smallest Superset

Theorem
$$\left({R \subseteq T}\right) \land \left({S \subseteq T}\right) \iff \left({R \cup S}\right) \subseteq T$$

Thus $$R \cup S$$ is the smallest subset of $$T$$ containing both $$R$$ and $$S$$, in the sense that for any $$A \subseteq T$$, if $$R \subseteq A$$ and $$S \subseteq A$$ then $$\left({R \cup S}\right) \subseteq A$$.

Proof

 * First we show $$\left({R \subseteq T}\right) \land \left({S \subseteq T}\right) \Longrightarrow \left({R \cup S}\right) \subseteq T$$:

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 * Next we show $$\left({R \cup S}\right) \subseteq T \Longrightarrow \left({R \subseteq T}\right) \land \left({S \subseteq T}\right)$$:

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Similarly for $$S$$:

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