Let $f_A:\emptyset\to A$ be the empty function with range $A$. The definition of a bijection as applied to this function is:
$$\forall x,y \in \emptyset (x=y \implies f_A(x)=f_A(y))$$
negating you get:
$$\exists x,y \in \emptyset (x = y\land f_A(x) \neq f_A(y))$$
Which is obviously a false statement since there are no elements in $\emptyset$ at all.
I got troubled by this question when considering the empty set as an inital object of the category Set and the following theorem:
"if I is an initial object then any object isomorphic to I is also an initial object."
but since every empty function is a bijection and thus an isomorphism it follows that all the objects in Set are initial which is obviously false.
What did i miss?