Proving axiom of incidence
These are the axioms of incidence from the book that I have:
I.1: Every line contains at least two distinct points.
I.2: There exists at least one line that contains two points.
I.3: There exists at most one line that containts two distinct points.
I.4: Every plane contains at least three non-colinear points.
I.5: There exists at least one plane that contains three points.
I.6: There exists at most one plane that contains three non-colinear points.
I.7: If two distinct points of some line belongs to one plane, then every point of that line belongs to same plane.
I.8: If two distinct planes have one common point, then they have at least one more common point.
I.9: There are four non-coplanar points.
I have to prove this:
"If we substitute axiom I.4 by axiom I.4': 'Every plane contains at least one point', then prove that consequence from given group of axioms (which I have typed) is axiom I.4"
My proof goes like this:
From I.9 we have three non-colinear points A,B,C which forms one plane by axiom I.6 and one point D not contained by plane .
Any two points of plane can form with point D another plane .
Any plane different then and must intersect one of them. By axiom I.4' that plane contains at least one point and by intersecting one of two planes or it has two more distinct points by axiom I.8 which means that every plane has three non-colinear points.
Is my proof correct?