*The comments by The User and Joel David Hamkins refer to a previous version of the answer which contained a mistake. The current version is completely disjoint of the previous one, and the comments no longer apply.*

This appears in Tarski's book **Cardinal Algebras** as Theorem 2.28, called *Interpolation Theorem*, and the statement of the theorem is as follows:

> If $n\leqq\infty,\ p\leqq\infty$, such that $a_i\leqq b_j$ for $i< n$ and $j < p$, then there is an element $c$ such that $a_i\leqq c\leqq b_j$ for every $i < n$ and $j < p$.

The theorem appears on page 27. The full citation is given below, one can also read about it [on MathSciNet](http://www.ams.org/mathscinet-getitem?mr=29954).

> Tarski, Alfred. **Cardinal Algebras. With an Appendix: Cardinal Products of Isomorphism Types, by Bjarni Jónsson and Alfred Tarski.** *Oxford University Press,* New York, N. Y., 1949. xii+326 pp.