If $ABC$ and $ABD$ are true and $C \neq D$, then either $ABCD$ or $ABDC$ is true. $ABC$ means $C$ is an element of $AB$ and $ABD$ means $D$ is an element of $AB$. Since $C \neq D$, then $A,$ $B,$ $C,$ and $D$ are collinear. If $A,$ $C,$ and $D$ are collinear and different points, then $CDA$, $DAC,$ or $ACD$ are true. If $CDA$ is true, then $ADC$ is true and $CAD$ is false. To prove $CDA$ is true, assume $CAD$ is true and deduce a contradiction. If $CAD$ is true, then $A$ is between point $C$ and $D,$ $B$ is between $A$ and $D.$ $B$ is also between $C$ and $A,$ here is the contradiction. Therefore $CDA$ is true and $DAC$ is false.
Am I on the right track here?