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	$$⊢P={\mathrm{Base}}_G$$
	tkgeom.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tkgeom.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tkgeom.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgbtwntriv2.1	$$⊢\phi \to A\in P$$
	tgbtwntriv2.2	$$⊢\phi \to B\in P$$
Assertion	tgbtwntriv1	$$⊢\phi \to A\in \left(AIB\right)$$

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	$$⊢P={\mathrm{Base}}_G$$
2		tkgeom.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tkgeom.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tkgeom.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		tgbtwntriv2.1	$$⊢\phi \to A\in P$$
6		tgbtwntriv2.2	$$⊢\phi \to B\in P$$
7	1 2 3 4 6 5	tgbtwntriv2	$$⊢\phi \to A\in \left(BIA\right)$$
8	1 2 3 4 6 5 5 7	tgbtwncom	$$⊢\phi \to A\in \left(AIB\right)$$