6. The number of elements in the set $\left\{(x, y, z) \left\lvert\, \log _{\frac{1}{4}}\left(x^{4}+y^{4}+z^{4}+1\right)\right.\right.$ $\left.\geqslant \log _{\frac{1}{4}} x+\log _{\frac{1}{4}} y+\log _{\frac{1}{4}} z-1\right\}$ is ( ).
(A) 0
(B) 1
(C) 2
(D) More than 2
6. (B).

From the known inequality, we get $x^{4}+y^{4}+z^{4}+1 \leqslant 4 x y z$ and $x, y, z$ are all positive numbers. Also, from the important inequality, we have $x^{4}+y^{4}+z^{4}+1 \geqslant 4 x y z$ (the equality holds if and only if $x=y=z=1$).
$$
\therefore x^{4}+y^{4}+z^{4}+1=4 x y z \text {. }
$$

Thus, $x=y=z=1$.