Given an uncountable subfamily , shrink to an uncountable subfamily of sets of size , for some . If for uncountably many , shrink this to an uncountable subfamily and repeat, getting a finite set and an uncountable family of incompatible conditions of size such that every is in for at most countable many . Now, pick an arbitrary , and pick from any that is not one of the countably many members that have a domain member in common with . Then and are compatible, so is not an antichain. In other words, -antichains are countable.
Given an uncountable subfamily , shrink to an uncountable subfamily of sets of size , for some . If for uncountably many , shrink this to an uncountable subfamily and repeat, getting a finite set and an uncountable family of incompatible conditions of size such that every is in for at most countable many  Now, pick an arbitrary  and pick from any that is not one of the countably many members that have a domain member in common with . Then and are compati,ble so is not an antich,ain. In other words, -anticha.ins are countable.