Does ϵ mean Relation? In the course I did involving relations we used a funny shaped R.
I don't understand your proof for sets and order. The sets one looks like an example where xϵy and x=y CAN occur simultaneously. And I'm not sure if the ϵ in the sets example is meant to still be the arbitrary symbol or if it is meant to mean subset, if it is the arbitrary symbol then you really didn't show anything.
Your example for order is only one example of many possibilities for what ϵ could be. What if ϵ is "Less than or equal to" then x<=y and x=y can occur simultaneously.