This follows from Ptolemy's inequality. Note that equality can't be achieved since no four points in a tetrahedron are coplanar.

Puzzled417 wrote: This follows from Ptolemy's inequality. Note that equality can't be achieved since no four points in a tetrahedron are coplanar. Yes, this is commonly known

Interestingly, there is an analogous proof of this to the 2-dimensional case (without reducing to the coplanar case). Ptolemy's inequality can be proven by inversion around one of the points and applying the triangle inequality. Now using sphere inversion, it's possible to show that the tetrahedron $ABCD$ has volume proportional to the area of a triangle with sides $AB\times CD,AC\times BD,AD\times BC$ (see the very end of the first answer to https://math.stackexchange.com/questions/1087011/calculating-the-radius-of-the-circumscribed-sphere-of-an-arbitrary-tetrahedron/2413725#2413725). In particular, Heron's formula would yield an imaginary volume for the tetrahedron of $ABCD$ if the problem statement were wrong, which is absurd.