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

*53. MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES.

Surmmary of *53.
The propositions to be given in this number are mostly such as would
have come more naturally at an earlier stage, but could not be given sooner
because they involved unit classes. It is to be observed that t'x u tiy is the
class consisting of the members x and y, while l'x tL'y is the relation which
holds only between x and y. If a and /3 are classes, t'a v t',3 is a class of
classes, its members being a and /3. If R and S are relations, t'R T i'S is a
relation of relations; and so on.
The present number begins by connecting products and sums p'K, s'K,
p'X, ^'X, in cases where the members of K or X are specified, with the products
or sums ar, a, a R i S, R S. We have
*53'01. F.p''a = a
*53-1. F.p'(t'a v t,/) = a e 13
*53'14. F.p'(c v t'a) =p'c n a
with similar propositions for s, p and s.
We have next a set of propositions on sums and products of classes of unit
classes. The most important of these is
*53'22. F. s't"a = a
We have next a proposition showing that the sum of K is null when, and
only when, K is either null or has the null-class for its only member, i.e.
*53'24. F:. s = A. -: K = A n Cls.v. K = tA
(Here we write "A n Cls," to show that the "A" in question is of the next
type above that of the other two A's.)
We have next various propositions on the relations of R'x and R'x and
R"a in various cases, first for a general relation R, and then for the particular
relation s defined in *40. Three of these propositions are very frequently
used, namely:
*533.:E! R'.-. R'xel
*53'301.. R"'x = R'x