An upper bound for union-closed family size

Let $\mathcal{A}$ be a union-closed family of sets with universe ${\bigcup }_{A\in \mathcal{A}}A=[n]=\{1,\cdots ,n\}$ and length $\mathrm{\ell }$. We prove that $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\le {\sum }_{i=0}^{\mathrm{\ell }}\left(\genfrac{}{}{0px}{}{n}{i}\right)$, with equality if and only if $\mathcal{A}={\bigcup }_{i=0}^{\mathrm{\ell }}\left(\genfrac{}{}{0px}{}{[n]}{n-i}\right)$. Additionally, by showing that $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\le \frac{{\mathrm{\ell }}^p-1}{\mathrm{\ell }-1}+2^n(1-2^{-\mathrm{\ell }}{)}^p$ for any nonnegative integer $p$, we establish for all integers $1\le k\le n$ that ${\sum }_{i=0}^k\left(\genfrac{}{}{0px}{}{n}{i}\right)\le \frac{k^{\widehat{p}}-1}{k-1}+2^n(1-2^{-k}{)}^{\widehat{p}}$, where $\widehat{p}=\lfloor (n-k)\mathrm{/}{\log }_2(\frac{k}{1-2^{-k}})\rfloor +1$.