Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

*84. CLASSES OF MUTUALLY EXCLUSIVE CLASSES.

Summary of *84.
A class K of mutuially exclusive classes is one such that, if a and f/ are
two different members of K, a and /3 have no common members; i.e. it is a
class composed of non-overlapping classes. Classes of mutually exclusive classes
have many important properties. They are important in cardinal arithmetic,
among other reasons, because if K is a class of mutually exclusive classes,
the cardinal number of s'K is the sum of the cardinal numbers of the
members of K. Also if K is a class of mutually exclusive classes, the number
of selected classes of K (i.e. D"e'CK) is the same as the number of selective
relations (i.e. i'Kc).
"K is a class of mutually exclusive classes" is written "K e Cls2 excl."
An important case is when no member of Ec is null; in this case we write
K e Cls ex2 excl.
For a Cls2 excl which is contained in a class of classes y, we write
C1 excl'y,
on the analogy of the notation Cl'y.
The definitions are as follows:
*84-01. Cls2excl = (a,, e K. a c.a,. a, a n = A) Df
*84-02. Cl excl'y = Cls2 excl n Cl'7 Df
*84-03. Cls ex2 excl = Cls2 excl - e 'A Df
The propositions of this number begin (*84'1 —14) with various equivalent
forms for the definitions. Of these the most useful are:
*84-11. F:. c e Cls2 excl.: a, 3e K.! a n. a,. a = /
*84'13. -: K e Cls ex2 excl. e Cls excl.
*84'14. F: Kc Cls2 excl. =. e e Cls- 1
The last of these is specially important, because it renders the propositions
of *81 applicable to efCK when K e Cls2 excl.