What happens if the empty set is not a subset of every set?
There is a logical error in your question. If $\emptyset=\{\}$, then, for any set $A$:
- It is true that every element of $\emptyset$ is in $A$, and
- It is true that every element of $\emptyset$ is not in $A$,
but
- It is not true that not every element of $\emptyset$ is in A. (Watch the word order!)
This is because $(\forall x)\neg P(x)$ is not equivalent to $\neg(\forall x)P(x)$ (i.e. universal quantifier does not swap with negation). Instead, it is equivalent to $\neg(\exists x)P(x)$ (the universal quantifier changes into existential quantifier).
(Note: In your case, $P(x) := x\in\emptyset\implies x\in A$.)
Here are some "unpleasant" consequences of your proposals.
Consider te "hypothesis":
the empty set is not a subset of any set.
This is: $\forall x \ \lnot ( \emptyset \subseteq x)$.
Applying the def of subset:
$\forall x \ \lnot \ \forall z \ (z \in \emptyset \to z \in x)$, i.e.
$\forall x \ \exists z \ \lnot (z \in \emptyset \to z \in x)$, i.e.
$\forall x \ \exists z \ (z \in \emptyset \land z \notin x)$.
This is true for every set $x$, and thus also of $\emptyset$:
$\exists z \ (z \in \emptyset \land z \notin \emptyset)$.
Consider now:
the empty set is not a subset of some sets.
This is: $\exists x \ \lnot ( \emptyset \subseteq x)$.
Applying the def of subset:
$\exists x \ \lnot \ \forall z \ (z \in \emptyset \to z \in x)$, i.e.
$\exists x \ \exists z \ (z \in \emptyset \land z \notin x)$.
No, it is not just a convention. And the statement "there exists a set $A$ with the property that not every element of $\{\}$ is an element of $A$" is never true.
Let $P$ be the property of being an element of $A$. Now every element of the empty set has property $P$ and not of property $P$, vacuously. The first part implies that the empty set is a subset of $A$ by the definition of being a subset.
But that every element of the empty set does not have property $P$ does not imply that the empty set is not a subset of $A$. The statement that the empty set is a subset of $A$ means that every element of the empty set (there are none) is an element of $A$. But that the empty set is not a subset of $A$ means that not all elements of it satisfy $P$. But if not all elements satisfy $A$, there must be an element that satisfies not $P$. In particular, there must be an element in the empty set- which is of course not true. All elements of the empty set satisfy not $P$, but that is not enough because there are no elements.