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

*90. ON THE ANCESTRAL RELATION.

Summary of *90.
If R is any relation, " xRy" is to mean "x is an ancestor of y with
respect to R," where a term counts as its own ancestor provided it belongs
to the field of R. The definition of RB is as follows:
90-01. R* = x {x C'R: R",u C,. x e. E. y e } Df
That is, xR*y is to hold when x belongs to the field of R, and y belongs to
every hereditary class to which x belongs; a hereditary class being a class /
such that R", C,a, i.e. such that all successors of it's are /'s.
*9002. RB = Cnv'R* Df
This definition serves merely to decide the ambiguity between (R)* and
Cnv'R*, either of which might be meant of RB. It will be shown, however,
that the two are equal (*90'132).
The most important propositions of this number are the following:
90'112. F:. xRsy: fz. zRw. Dz, w. w: x:D. oy
I.e. if xR*y and if Oz is a hereditary property belonging to x, then it
belongs to y.
*90-12. F: e C'R. xRx
I.e. RB is reflexive throughout the field of R, but not elsewhere.
*90'14.. D'R = ('R = C'R, = C'R
*90-15. F.I C'R C R
*90-151.. R C R
*9016. F.R,* R R
*90-163. F. R"R*'x C R,'x
I.e. R*'x is a hereditary class.
*90-17. F. R2= R*