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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	btwnlng1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	btwnlng1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
	btwnlng1.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
	btwnlng1.d	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
	btwnlng1.1	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) )
Assertion	btwnlng1	⊢ ( 𝜑 → 𝑍 ∈ ( 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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		btwnlng1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
6		btwnlng1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
7		btwnlng1.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
8		btwnlng1.d	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
9		btwnlng1.1	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) )
10	9	3mix1d	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) )
11	1 3 2 4 5 6 8 7	tgellng	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) )
12	10 11	mpbird	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) )