Consider a set $A$, elements $x,y$ in $A$ and the following propositions: \begin{equation} \exists x\in A\ |\quad x=x \end{equation} \begin{equation} \forall x\in A:\quad x=x \end{equation} \begin{equation} \forall x\in A,\ \exists y\in A\ |\quad x=y \end{equation} \begin{equation} \exists x\in A\ |\ \forall y\in A:\quad x=y \end{equation} The first three conditions are true for every set, while the last condition implies that $A$, if non empty, has only one element.
My question is this: are there any differences whatsoever between the first three propositions or are they simply interchangheable?
Only the 2nd and 3rd propositions are equivalent, because they are both universally true, both because of the reflexive property of equivalence, that always $z = z$:
$$\forall x \in A ~ x = x \tag{2}$$ $$\forall x \in A ~ \top$$ $$\top$$
$$\forall x \in A ~\exists y \in A ~ x = y \tag{3}$$ $$\forall x \in A ~ x = x\top$$ $$\forall x \in A ~ \top$$ $$\top$$
The first statement is equivalent to saying that $A$ is nonempty because $\exists x \in A ~ \top$ can only instantiate the $x$ from an element in $A$.
The last statement is equivalent to stating that $A$ has exactly 1 element.