I am trying to understand the difference between and $\wedge$ and $\rightarrow$. Consider $$x=y\Longleftrightarrow \forall z(z\in x \ \rightarrow z \in y)$$
However, I think we can use and notation rather than implies $$x=y\Longleftrightarrow \forall z(z\in x \ \wedge \ z \in y)$$
Can you explain briefly we we should use implies notation there?
In English, the first is read as "for all $z$, $z$ being in $x$ implies that $z$ is in $y$". This means that everything in $x$ is also in $y$.
This second is read as "for all $z$, $z$ is in $x$ and $z$ is in $y$". This means that both $x$ and $y$ contain everything.
It is worth noting that neither of the two statements above are equivalent to $x=y$, so there is no reason to use $\implies$ instead of $\land$ in this case; they just mean different things.
