Question

Complete this formal proof of Ex(P(x)v~P(x)) from the empty set. NOTE: similar to the rule above when instantiating quantifiers, if you need a random name, always start at the beginning of the alphabet. That is, use a first; only use b if necessary; etc.

Answer 1

Some proofs of definitions by contradiction and contrapositive but not direct proofs (the existence of infinite primes for example),
I think most of them because the direct proof extends away from a first mathematics course or the proofs by contradiction/contrapositive are more didactic.
The one that most bothers me in particular is the demonstration that the empty set is a subset of every set, and it is unique.
I understand the uniqueness and understand the proof by contradiction:
Suppose ∅⊊A where A is a set. So it exists an element x∈∅ such that x∉A wich is absurd because ∅ does not have any elements by definition.
but I would like to know if there exists a direct proof of this and if indeed extends from a first course.