I want to show that the intersection of any inductive set is empty since every inductive set contains the empty set.
I thought that we could do it like that:
We know that $B$ is an inductive set. So:
$$\varnothing \in B \wedge \forall x(x \in B \rightarrow x' \in B)$$
$$y \in \bigcap B \leftrightarrow \forall b \in B: y \in b$$
Since $\varnothing \in B$ we get that $y \in \bigcap B \leftrightarrow y \in \varnothing$.
But since there is no $y$ such that $y \in \varnothing$ we conclude that we cannot find a $y$ such that $y \in \bigcap B$.
But it isn't right, since we cannot just take one set to get the equivalence, right?
How else could we do this?