Equivalence relations on classes instead of sets

Can someone please explain to me how to deal with equivalence relations on classes instead of sets? Is there some sort of generalisation of relations?

Thank you

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It’s not clear to me whether you’re asking what a class equivalence relation would look like formally, or how to work with one when you have it. I’ve given an answer based on the first interpretation, and Asaf has sketched an answer to the second interpretation.

Suppose that a class $\mathbf C$ is described by a formula $\varphi$: $x\in\mathbf{C}\leftrightarrow\varphi(x)$. A formula $\psi(x,y)$ describes a class equivalence relation on $\mathbf C$ if it satisfies the following conditions:

• $\forall x\Big(\varphi(x)\to\psi(x,x)\Big)$ (reflexivity)
• $\forall x,y\Big(\varphi(x)\land\varphi(y)\land\psi(x,y)\to\psi(y,x)\Big)$ (symmetry)
• $\forall x,y,z\Big(\varphi(x)\land\varphi(y)\land\varphi(z)\land\psi(x,y)\land\psi(y,z)\to\psi(x,z)\Big)$ (transitivity)
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No fair I am writing from an iPhone. Also we read the question in two very different ways. –  Asaf Karagila Dec 6 '12 at 1:12
@Asaf: Your interpretation may be right; I really can’t tell. You might want to add a link for Scott’s trick. –  Brian M. Scott Dec 6 '12 at 1:16
Well. I hope to find out in six hours. Now I have an appointment with the Sandman. Also could you please edit the link in? Doing that from the iPhone is a true pain in one lower back. –  Asaf Karagila Dec 6 '12 at 1:17