Let $(\Gamma,\leq)$ be a complete lattice and put $\wedge_{x\in \Gamma}x =0$. Assume $A$ and $B$ are two subsets of $\Gamma$  with $a\wedge b=0$ for every $a\in A$ and $b\in B$.

Q. True or false:
 $$\wedge\{ a\vee b : a\in A , b\in B\}\leq (\wedge_{a\in A}a)\vee(\wedge_{b\in B}b) $$