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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwntriv2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwntriv2.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
Assertion	tgbtwntriv1	⊢ ( 𝜑 → 𝐴 ∈ ( 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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgbtwntriv2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgbtwntriv2.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7	1 2 3 4 6 5	tgbtwntriv2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐴 ) )
8	1 2 3 4 6 5 5 7	tgbtwncom	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 𝐼 𝐵 ) )