Labelling classes by sets

Let Q be an equivalence relation whose equivalence classes, denoted Q [ x ], may be proper classes. A function L defined on Field( Q ) is a labelling for Q if and only if for all x, L ( x ) is a set and L is a labelling by subsets for Q if and only if BG denotes Bernays-GOdel class-set theory with neither the axiom of foundation, AF, nor the class axiom of choice, E. The following are relatively consistent with BG. (1) E is true but there is an equivalence relation with no labelling. (2) E is true and every equivalence relation has a labelling, but there is an equivalence relation with no labelling by subsets. [PUBLICATION ABSTRACT]