source : https://www.cse.iitb.ac.in/~nutan/courses/cs207-12/notes/lec7.pdf
If the longest chain in a partial order is of length n, then the partial order can be written as a Partition of n antichains.
https://www.cse.iitb.ac.in/~nutan/courses/cs207-12/notes/lec7.pdf
Reference for more details.
Just extra information about What is antichain? A chain in S is a subset C of S in which each pair of elements is comparable; that is, C is totally ordered. An antichain in S is a subset A of S in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in A. Reference:https://en.wikipedia.org/wiki/Antichain
CHAIN : Let (A ,⋨) be a Poset .A subset of A is called Chain if every two elements in the subset are related.
ANTICHAIN: A subset of A is called Antichain if no two distinct elements in the subset are related.
THEOREM: Let (A ,⋨) be a Poset.Suppose the length of the longest chains in A is n.Then the elements in A can be partitioned into n disjoint Antichains.
@Vicky rix Won't {{1}} and {{2}} also be antichains, in addition to {{1},{2}}.
Edit: Found this source http://mathworld.wolfram.com/Antichain.html