Theorem tgbtwntriv1

Description: Betweenness always holds for the first endpoint. Theorem 3.3 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019)

	Ref	Expression
Hypotheses	tkgeom.p	|- P = ( Base ` G )
	tkgeom.d	|- .- = ( dist ` G )
	tkgeom.i	|- I = ( Itv ` G )
	tkgeom.g	|- ( ph -> G e. TarskiG )
	tgbtwntriv2.1	|- ( ph -> A e. P )
	tgbtwntriv2.2	|- ( ph -> B e. P )
Assertion	tgbtwntriv1	|- ( ph -> A e. ( A I B ) )

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	|- P = ( Base ` G )
2		tkgeom.d	|- .- = ( dist ` G )
3		tkgeom.i	|- I = ( Itv ` G )
4		tkgeom.g	|- ( ph -> G e. TarskiG )
5		tgbtwntriv2.1	|- ( ph -> A e. P )
6		tgbtwntriv2.2	|- ( ph -> B e. P )
7	1 2 3 4 6 5	tgbtwntriv2	|- ( ph -> A e. ( B I A ) )
8	1 2 3 4 6 5 5 7	tgbtwncom	|- ( ph -> A e. ( A I B ) )