I'm reading: Devlin's Joy of Sets.
He gives the definition of the axiom of extensionality:
The definition of subset:
And then there is this exercise:
Rewriting it, I'd have:
\begin{eqnarray*} {(x=y)}& \iff &{(x\subseteq y)\wedge (y\subseteq x)} \\ {}&\iff &{\forall a[a\in x\implies a\in y] \wedge \forall a[a\in y\implies a\in x]} \end{eqnarray*}
Are the following two expressions equivalent:
$$\forall a[a\in x\implies a\in y] \wedge \forall a[a\in y\implies a\in x] \\ \stackrel{?}{\equiv}\\ \forall a([a\in x\implies a\in y] \wedge [a\in y\implies a\in x])$$
Intuitively, I guess they are. But I'm not sure.
