Is "The empty set is a subset of any set" a convention? Recently I learned that for any set A, we have $\varnothing\subset A$.

I found some explanation of why it holds.


  $\varnothing\subset A$ 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 $\varnothing\subset A$ 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 $\varnothing\not\subset A$ 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 $B=\varnothing$.
I'm also learning about sets, but the way that I've begun to see things like this is as follows:

I think about what a nice system needs, and a few things spring to mind. I need an addition, the idea of pushing two things together. How about a concept of negatives? That could be useful. I could add these negatives to some of these positives, and eventually I'm going to come across a situation where my negatives meets my positives and what do I have? That's the null set.

Think about 'zero' logically, this concept of nullness, but more than just the number zero, but its function, and realize why it fits soundly within our understanding of mathematics. Say we were to not have this zero, the what would happen? This concept of zero is such an obvious idea in numbers that when we get zero in more bizarre systems like sets, the concept of zero along this feels odd, because 0 is just a number after all.

But in sets your '0' still retains the set construct, but it represents the analogous function that 0 holds in numbers.