Recently I learned that for any set A, we have ∅⊂A.
I found some explanation of why it holds.
∅⊂A means “for every object x, if x belongs to the empty set, then x also belongs to the set A”. This is a vacuous truth, because the antecedent (x belongs to the empty set) could never be true, so the conclusion always holds (x also belongs to the set A). So ∅⊂A holds.
What confused me was that, the following expression was also a vacuous truth.
For every object x, if x belongs to the empty set, then x doesn’t belong to the set A.
According to the definition of the vacuous truth, the conclusion (x doesn’t belong to the set A) holds, so ∅⊄ would be true, too.
Which one is correct? Or is it just a convention to let \varnothing\subset A?
There’s no conflict: you’ve misinterpreted the second highlighted statement. What it actually says is that \varnothing and A have no element in common, i.e., that \varnothing\cap A=\varnothing. This is not the same as saying that \varnothing is not a subset of A, so it does not conflict with the fact that \varnothing\subseteq A.
To expand on that a little, the statement B\nsubseteq A does not say that if x\in B, then x\notin A; it says that there is at least one x\in B that is not in A. This is certainly not true if B=\varnothing.