Variant implication introduction rules

With $(\vee I)$, we can get from $\neg A$ to $\neg A\vee B$ and $B$ to $\neg A\vee B$, which we know are equivalent to $A\to B$. Using $(\to I1)$ we can derive an implication whose antecedent is false, and using $(\to I2)$ we can derive an implication whose consequence is true: ?

$$\frac{\neg A}{A\to B}(\to I1)\frac{B}{A\to B}(\to I2)$$