I'm studying axiomatic Euclidean Geometry, and I'm using two books which have slightly different set of axioms. Here's the set of Incidence Axioms and Betweenness Axioms of the first one:
I1) For any line, there are at least two points in the line and two points outside of it.
I2) For any given two distinct points, there is a unique line that contains both of them.
B1) For any given three distinct points in a line, one and only one is between the other two.
B2) For any two distinct points, there are points $C$ and $D$ such that $C$ is between $A$ and $B$, and $B$ is between $A$ and $D$.
B3) A line determines exactly two semiplanes.
Where, by definition, given a line $l$ and a point $A$, the semiplane determined by $l$ and $A$ is the set of points $B$ such that $AB$ (the set of points between $A$ and $B$) does not intersect the line l.
Using that, I'd like to prove Pasch's Axiom:
"Let $A$, $B$, $C$ be three points that do not lie on a line and let $l$ be a line which does not meet any of the points $A$, $B$, $C$. If the line $l$ passes through a point of the segment $AB$, it also passes through a point of the segment $AC$, or through a point of segment $BC$."
Is it possible? I'm not sure if it's provable without using the later axioms of the book.