Intersection with Subset is Subset

Theorem

 * $$S \subseteq T \iff S \cap T = S$$

where:
 * $$S \subseteq T$$ denotes that $$S$$ is a subset of $$T$$;
 * $$S \cap T$$ denotes the intersection of $$S$$ and $$T$$.

Proof

 * Let $$S \cap T = S$$.

Then by the definition of set equality, $$S \subseteq S \cap T$$.

Thus:

$$ $$

Now let $$S \subseteq T$$.

From Intersection Subset we have $$S \supseteq S \cap T$$.

We also have:

$$ $$ $$

So as we have:

$$ $$

it follows from the definition of Set Equality that we have $$S \cap T = S$$.

So we have:

$$ $$

and so:
 * $$S \subseteq T \iff S \cap T = S$$

from the definition of equivalence.