Theorem btwncolg1

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

	Ref	Expression
Hypotheses	tglngval.p	$$⊢P={\mathrm{Base}}_G$$
	tglngval.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	tglngval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tglngval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tglngval.x	$$⊢\phi \to X\in P$$
	tglngval.y	$$⊢\phi \to Y\in P$$
	tgcolg.z	$$⊢\phi \to Z\in P$$
	btwncolg1.z	$$⊢\phi \to Z\in \left(XIY\right)$$
Assertion	btwncolg1	$$⊢\phi \to \left(Z\in \left(XLY\right)\vee X=Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		tglngval.p	$$⊢P={\mathrm{Base}}_G$$
2		tglngval.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
3		tglngval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tglngval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		tglngval.x	$$⊢\phi \to X\in P$$
6		tglngval.y	$$⊢\phi \to Y\in P$$
7		tgcolg.z	$$⊢\phi \to Z\in P$$
8		btwncolg1.z	$$⊢\phi \to Z\in \left(XIY\right)$$
9	8	3mix1d	$$⊢\phi \to \left(Z\in \left(XIY\right)\vee X\in \left(ZIY\right)\vee Y\in \left(XIZ\right)\right)$$
10	1 2 3 4 5 6 7	tgcolg	$$⊢\phi \to \left(\left(Z\in \left(XLY\right)\vee X=Y\right)\leftrightarrow \left(Z\in \left(XIY\right)\vee X\in \left(ZIY\right)\vee Y\in \left(XIZ\right)\right)\right)$$
11	9 10	mpbird	$$⊢\phi \to \left(Z\in \left(XLY\right)\vee X=Y\right)$$