Theorem tgbtwnne

Description: Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-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	$$⊢\phi \to B\in P$$
	tgbtwncomb.3	$$⊢\phi \to C\in P$$
	tgbtwnne.1	$$⊢\phi \to B\in \left(AIC\right)$$
	tgbtwnne.2	$$⊢\phi \to B\ne A$$
Assertion	tgbtwnne	$$⊢\phi \to A\ne C$$

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		tgbtwncomb.3	$$⊢\phi \to C\in P$$
8		tgbtwnne.1	$$⊢\phi \to B\in \left(AIC\right)$$
9		tgbtwnne.2	$$⊢\phi \to B\ne A$$
10	4	adantr	$$⊢\left(\phi \wedge A=C\right)\to G\in {𝒢}_{\mathrm{Tarski}}$$
11	5	adantr	$$⊢\left(\phi \wedge A=C\right)\to A\in P$$
12	6	adantr	$$⊢\left(\phi \wedge A=C\right)\to B\in P$$
13	8	adantr	$$⊢\left(\phi \wedge A=C\right)\to B\in \left(AIC\right)$$
14		simpr	$$⊢\left(\phi \wedge A=C\right)\to A=C$$
15	14	oveq2d	$$⊢\left(\phi \wedge A=C\right)\to AIA=AIC$$
16	13 15	eleqtrrd	$$⊢\left(\phi \wedge A=C\right)\to B\in \left(AIA\right)$$
17	1 2 3 10 11 12 16	axtgbtwnid	$$⊢\left(\phi \wedge A=C\right)\to A=B$$
18	17	eqcomd	$$⊢\left(\phi \wedge A=C\right)\to B=A$$
19	9	adantr	$$⊢\left(\phi \wedge A=C\right)\to B\ne A$$
20	19	neneqd	$$⊢\left(\phi \wedge A=C\right)\to \neg B=A$$
21	18 20	pm2.65da	$$⊢\phi \to \neg A=C$$
22	21	neqned	$$⊢\phi \to A\ne C$$