Cardinality of compacts with positive $(>0)$ Lebesgue measure in $\Bbb R^3$

Question

I need to prove that it's same as $[0,1]$ (continuum).

Let's say I have proved "fact" that closed balls with positive radius in $\Bbb R^3$ have same cardinality as $[0,1]$.

Does it prove that compacts have cardinality no more than continuum? If so, what I should do in "no less than" part (or vice versa no less than -> no more than)?