Why does every x belong to A in the intersection of an empty collection?

[micromass]

I disagree on the following point.

I overlooked that you are referring to: y∈{x | φ(x)} ↔ φ(y) as though it is a definition, when it is actually not.

y∈{x | φ(x)} ↔ φ(y) is a theorem of naive set theory that restates the axiom of abstraction using the {...|...} operator.

In ZF, this theorem admits Russell's paradox, and cannot be a theorem of ZF.

The correct theorem in ZF is:

$$y \in \{x | \phi (x) \} \Rightarrow \phi (y)$$

The equivalence cannot be justified from the separation axiom, and must be an implication to avoid the paradox.

And as you may probably have already guessed, and operation definition for {x | φ(x)} is required in order to eliminate {x | φ(x)} entirely to wff's in the object language:

$$\{x | \phi (x) \} = w \Leftrightarrow [ \forall x (x \in w \Leftrightarrow \phi (x)) \wedge w \mbox{ is a set} ] \vee [\neg \exists B \forall x (x \in B \Leftrightarrow \phi (x)) \wedge w = \emptyset]$$

This definition along with

$$\{ x | \phi (x) \} \not = \emptyset$$

proves the implication, but the converse


$$\phi (y) \Rightarrow y \in \{x | \phi (x) \}$$

cannot be shown to follow from separation.

Yes, I understand you completely. But I think you fail to understand me. You're taking the point-of-view of Suppes. This is not standard terminology!
Please, read another book on set theory and you will see what I'm talking about.

I can make suggestion to you that will increase your understanding and point-of-views. But only you can follow them

Good luck on your research!