Theorem tgbtwnne

Description: Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019)

	Ref	Expression
Hypotheses	tkgeom.p	|- P = ( Base ` G )
	tkgeom.d	|- .- = ( dist ` G )
	tkgeom.i	|- I = ( Itv ` G )
	tkgeom.g	|- ( ph -> G e. TarskiG )
	tgbtwntriv2.1	|- ( ph -> A e. P )
	tgbtwntriv2.2	|- ( ph -> B e. P )
	tgbtwncomb.3	|- ( ph -> C e. P )
	tgbtwnne.1	|- ( ph -> B e. ( A I C ) )
	tgbtwnne.2	|- ( ph -> B =/= A )
Assertion	tgbtwnne	|- ( ph -> A =/= C )

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	|- ( ph -> G e. TarskiG )
5		tgbtwntriv2.1	|- ( ph -> A e. P )
6		tgbtwntriv2.2	|- ( ph -> B e. P )
7		tgbtwncomb.3	|- ( ph -> C e. P )
8		tgbtwnne.1	|- ( ph -> B e. ( A I C ) )
9		tgbtwnne.2	|- ( ph -> B =/= A )
10	4	adantr	|- ( ( ph /\ A = C ) -> G e. TarskiG )
11	5	adantr	|- ( ( ph /\ A = C ) -> A e. P )
12	6	adantr	|- ( ( ph /\ A = C ) -> B e. P )
13	8	adantr	|- ( ( ph /\ A = C ) -> B e. ( A I C ) )
14		simpr	|- ( ( ph /\ A = C ) -> A = C )
15	14	oveq2d	|- ( ( ph /\ A = C ) -> ( A I A ) = ( A I C ) )
16	13 15	eleqtrrd	|- ( ( ph /\ A = C ) -> B e. ( A I A ) )
17	1 2 3 10 11 12 16	axtgbtwnid	|- ( ( ph /\ A = C ) -> A = B )
18	17	eqcomd	|- ( ( ph /\ A = C ) -> B = A )
19	9	adantr	|- ( ( ph /\ A = C ) -> B =/= A )
20	19	neneqd	|- ( ( ph /\ A = C ) -> -. B = A )
21	18 20	pm2.65da	|- ( ph -> -. A = C )
22	21	neqned	|- ( ph -> A =/= C )