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.
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.