Subset Equivalences

Theorem
These statements are equivalent:


 * $$S \subseteq T$$
 * $$S \cup T = T$$

(work in progress)

Proof
Let $$S \cup T = T$$.

From Subset Union, $$S \subseteq S \cup T$$

Hence from Subsets Transitive, $$S \subseteq T$$.

Thus $$S \cup T = T \Longrightarrow S \subseteq T$$.

Now let $$S \subseteq T$$.

From Subset of Itself, $$T \subseteq T$$.

By the Rule of Conjunction, $$S \subseteq T \land T \subseteq T$$.

Thus from Smallest Union, $$S \cup T \subseteq T$$

From Subset Union, we have $$T \subseteq S \cup T$$.

So we have $$T \subseteq S \cup T \land S \cup T \subseteq T$$.

Hence by the definition of set equality, $$S \cup T = T$$.

Thus $$S \subseteq T \Longrightarrow S \cup T = T$$.

Hence $$S \subseteq T \iff S \cup T = T$$, so $$S \cup T = T$$ and $$S \subseteq T$$ are equivalent.