Union is Smallest Superset

Theorem
Let $$S_1$$ and $$S_2$$ be sets.

Then $$S_1 \cup S_2$$ is the smallest set contained in both $$S_1$$ and $$S_2$$.

That is:
 * $$\left({S_1 \subseteq T}\right) \and \left({S_2 \subseteq T}\right) \iff \left({S_1 \cup S_2}\right) \subseteq T$$

Generalized Result
Let $$S_i \subseteq S: i \in \N^*_n$$.

Then:
 * $$\left({\forall i \in \N^*_n: S_i \subseteq T}\right) \iff \bigcup_{i = 1}^n S_i \subseteq T$$.

Proof

 * Let $$\left({S_1 \subseteq T}\right) \and \left({S_2 \subseteq T}\right)$$.

Then:

$$ $$ $$

So:
 * $$\left({S_1 \subseteq T}\right) \and \left({S_2 \subseteq T}\right) \implies \left({S_1 \cup S_2}\right) \subseteq T$$.


 * Next we show $$\left({S_1 \cup S_2}\right) \subseteq T \implies \left({S_1 \subseteq T}\right) \and \left({S_2 \subseteq T}\right)$$:

$$ $$ $$

Similarly for $$S$$:

$$ $$ $$


 * So, from the above, we have:


 * 1) $$\left({S_1 \subseteq T}\right) \and \left({S_2 \subseteq T}\right) \implies \left({S_1 \cup S_2}\right) \subseteq T$$;
 * 2) $$\left({S_1 \cup S_2}\right) \subseteq T \implies \left({S_1 \subseteq T}\right) \and \left({S_2 \subseteq T}\right)$$.

Thus $$\left({S_1 \subseteq T}\right) \and \left({S_2 \subseteq T}\right) \iff \left({S_1 \cup S_2}\right) \subseteq T$$ from the definition of equivalence.

Generalized Proof
Proof by induction:

For all $$n \in \N^*$$, let $$P \left({n}\right)$$ be the proposition:
 * $$\left({\forall i \in \N^*_n: S_i \subseteq T}\right) \iff \bigcup_{i = 1}^n S_i \subseteq T$$.

$$P(1)$$ is trivially true, as this just says $$S_1 \subseteq T \iff S_1 \subseteq T$$.

Basis for the Induction
$$P(2)$$ is the case:
 * $$\left({S_1 \subseteq T}\right) \and \left({S_2 \subseteq T}\right) \iff \left({S_1 \cup S_2}\right) \subseteq T$$

which has been proved above.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $$P \left({k}\right)$$ is true, where $$k \ge 2$$, then it logically follows that $$P \left({k+1}\right)$$ is true.

So this is our induction hypothesis:


 * $$\left({\forall i \in \N^*_k: S_i \subseteq T}\right) \iff \bigcup_{i = 1}^k S_i \subseteq T$$.

Then we need to show:


 * $$\left({\forall i \in \N^*_{k+1}: S_i \subseteq T}\right) \iff \bigcup_{i = 1}^{k+1} S_i \subseteq T$$.

Induction Step
This is our induction step:

$$ $$ $$ $$

So $$P \left({k}\right) \implies P \left({k+1}\right)$$ and the result follows by the Principle of Mathematical Induction.

Therefore $$\left({\forall i \in \N^*_n: S_i \subseteq T}\right) \iff \bigcup_{i = 1}^n S_i \subseteq T$$.