Question

What is the meaning of "there exists x ∈ S ∖ m. Then m ⊊ m ∪ {x} ∈ X; a contradiction"?

Answer 1

It means that there is an element x in the set S that is not in the set m. Then, m is a proper subset of m union {x}, which is a member of the set X. This leads to a contradiction, which means that the original assumption must be false.