Definition of "antichain"

First of all, Zorn's Lemma implies that every uncountable antichain in T is contained in a maximal uncountable antichain. So it suffices to make sure that T has no maximal uncountable antichains.

1997, Winfried Just, Martin Weese, Discovering Modern Set Theory II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 140

We first define three different but isomorphic lattices: the lattice of maximal antichain ideals, the lattice of maximal antichains and the lattice of strict ideals.

2013, Vijay K. Garg, Maximal Antichain Lattice Algorithms for Distributed Computations, Proceedings, Davide Frey, Michel Raynal, Saswati Sarkar, Rudrapatna K. Shyamasundar, Prasun Sinha (editors), Distributed Computing and Networking: 14th International Conference, ICDCN, Springer, LNCS 7730, page 245

In 1928 Emanuel Sperner asked and answered the following question: Suppose we are given the set N = { 1 , 2 , 3 , … , n } {\displaystyle N=\{1,2,3,\dots ,n\}} . Call a family F {\displaystyle {\mathcal {F}}} of subsets of N {\displaystyle N} [i.e., F ⊆ the power set P(N), which has partial order ⊆] an antichain if no set of F {\displaystyle {\mathcal {F}}} contains another set of the family F {\displaystyle {\mathcal {F}}} . What is the size of a largest antichain? Clearly, the family F k {\displaystyle {\mathcal {F}}_{k}} of all k {\displaystyle k} -sets satisfies the antichain property with | F k | = ( n k ) {\displaystyle \textstyle |{\mathcal {F}}_{k}|={\binom {n}{k}}} . Looking at the maximum of the binomial coefficients (see page 14) we conclude that there is an antichain of size ( n [ n / 2 ] ) = max k ( n k ) {\displaystyle \textstyle {\binom {n}{[n/2]}}=\max _{k}{\binom {n}{k}}} . Sperner's theorem now asserts that there are no larger ones.

2014, Martin Aigner, Günter M. Ziegler, Proofs from THE BOOK, Springer, 5th Edition, page 199