Alternatives to Fano's Axiom in Projective Space In Projective Geometry, Fano's Axiom says:

The three diagonal points of a complete quadrangle are never collinear.

I would like to prove this from more basic Axioms within three-dimensional Projective Space. The theorem of Desargues is non-trivial in plane geometry, but can be proven from basic axioms within three-dimensional geometry; could the same be true for Fano's Axiom? If not, are there nice (equivalent) alternatives?
Edit: by “more basic” I mean intuitively more basic, which makes it a somewhat subjective question of course.

You can't prove Fano's axiom from 3-dimensional geometry because the projective plane over the field $F_2$ with two elements does not satisfy Fano's axioms.  Recall that the projective plane can be defined starting with 3-dimensional space over $F_2$ by a suitable equivalence relation.
