How to demonstrate the subset definition?

Question

I'm having a hard time trying to demonstrate the definition of a subset, which is :

$S \subseteq T \Leftrightarrow (S \subset T) \vee (S = T)$

I always end up by expending the definitions of $\subset$ and $=$ in Boolean algebra, resulting into this complicated situation.

Any way to solve it by staying in the Set theory? Or in Boolean algebra?

Answer 1

You're on the right way!

Your definition of $S \subset T$ as $\forall x (x \in S \rightarrow x \in T) \land \exists x (x \in T \land \neg x \in S)$ should work fine.

For $S = T$ you can use $\forall x (x \in S \leftrightarrow x \in T)$

And for $S \subseteq T$ you can use $\forall x (x \in S \rightarrow x \in T)$

Using these definitions, you should be able to prove $S \subseteq T \leftrightarrow (S \subset T \lor S =T)$ just fine.