Upward Lowenheim-Skolem Theorem: For any infinite $L$-structure $\mathfrak{A}$ and for any cardinal $\kappa\geq|A|+|L|$, there is a structure $\mathfrak{B}$ such that $\mathfrak{A}\preceq\mathfrak{B}$ and $|B|=\kappa$.
Each roman letter denotes the domain of the structure represented by the corresponding gothic form. For example, dom$(\mathfrak{A})=A$.
As a corollary: If $T$ is an $L$-theory and $T$ has an infinite model, then for any cardinal $\kappa\geq\aleph_0+|L|$, $T$ has a model of cardinality $\kappa$.
But why can you replace $|A|$ by $\aleph_0$ in the corollary? What if $|A|$ has a greater cardinality than $\aleph_0$?