Literature about a property of union closed families?

This is similar to your proof but without induction.

We prove that there are at least 3 such sets. For $r=k$ this is clear, so assume that $k>r$. Consider our $\binom{k}{r}+1$ $r$-sets. Call an element $v\in V$ appropriate if $v$ belongs to at most $\binom{k-1}{r-1}$ our sets. Then there exist at least $\binom{k}{r}+1-\binom{k-1}{r-1}=\binom{k-1}{r}+1$ our sets not containing $v$. Their union contains at least $k$ elements, and does not contain $v$. Now I claim that between any $k$ elements $x_1,\dots,x_k$ there exists an appropriate element $v$. Indeed, if not, then total number of pairs (our $r$-set $A$, $x_i\in A$) is at least $k(\binom{k-1}{r-1}+1)>r (\binom{k}{r}+1)$, a contradiction. So, we may find appropriate element $v$, the union $U$ of our sets not containing $v$ has cardinality at least $k$. Thus there exists appropriate $u\in U$ and the union of our sets which do not contain $u$ is a third set after $V,U$.

I wonder whether bound 3 may be further improved (for some values of $k,r$, of course for $k=r$ it can not.)