Theorem btwnlng1

Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019)

	Ref	Expression
Hypotheses	btwnlng1.p	|- P = ( Base ` G )
	btwnlng1.i	|- I = ( Itv ` G )
	btwnlng1.l	|- L = ( LineG ` G )
	btwnlng1.g	|- ( ph -> G e. TarskiG )
	btwnlng1.x	|- ( ph -> X e. P )
	btwnlng1.y	|- ( ph -> Y e. P )
	btwnlng1.z	|- ( ph -> Z e. P )
	btwnlng1.d	|- ( ph -> X =/= Y )
	btwnlng1.1	|- ( ph -> Z e. ( X I Y ) )
Assertion	btwnlng1	|- ( ph -> Z e. ( X L Y ) )

Proof

Step	Hyp	Ref	Expression
1		btwnlng1.p	|- P = ( Base ` G )
2		btwnlng1.i	|- I = ( Itv ` G )
3		btwnlng1.l	|- L = ( LineG ` G )
4		btwnlng1.g	|- ( ph -> G e. TarskiG )
5		btwnlng1.x	|- ( ph -> X e. P )
6		btwnlng1.y	|- ( ph -> Y e. P )
7		btwnlng1.z	|- ( ph -> Z e. P )
8		btwnlng1.d	|- ( ph -> X =/= Y )
9		btwnlng1.1	|- ( ph -> Z e. ( X I Y ) )
10	9	3mix1d	|- ( ph -> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) )
11	1 3 2 4 5 6 8 7	tgellng	|- ( ph -> ( Z e. ( X L Y ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )
12	10 11	mpbird	|- ( ph -> Z e. ( X L Y ) )