Theorem tgbtwnconn2

Description: Another connectivity law for betweenness. Theorem 5.2 of Schwabhauser p. 41. (Contributed by Thierry Arnoux, 17-May-2019)

	Ref	Expression
Hypotheses	tgbtwnconn.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tgbtwnconn.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tgbtwnconn.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwnconn.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwnconn.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgbtwnconn.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgbtwnconn.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	tgbtwnconn2.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
	tgbtwnconn2.2	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
	tgbtwnconn2.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
Assertion	tgbtwnconn2	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ∨ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tgbtwnconn.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tgbtwnconn.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		tgbtwnconn.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
4		tgbtwnconn.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
5		tgbtwnconn.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
6		tgbtwnconn.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
7		tgbtwnconn.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
8		tgbtwnconn2.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
9		tgbtwnconn2.2	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
10		tgbtwnconn2.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
11		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
12	3	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐺 ∈ TarskiG )
13	4	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐴 ∈ 𝑃 )
14	5	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐵 ∈ 𝑃 )
15	6	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐶 ∈ 𝑃 )
16	7	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐷 ∈ 𝑃 )
17	9	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
18		simpr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) )
19	1 11 2 12 13 14 15 16 17 18	tgbtwnexch3	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) )
20	19	orcd	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → ( 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ∨ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) )
21	3	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐺 ∈ TarskiG )
22	4	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐴 ∈ 𝑃 )
23	5	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐵 ∈ 𝑃 )
24	7	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐷 ∈ 𝑃 )
25	6	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐶 ∈ 𝑃 )
26	10	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
27		simpr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) )
28	1 11 2 21 22 23 24 25 26 27	tgbtwnexch3	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) )
29	28	olcd	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → ( 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ∨ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) )
30	1 2 3 4 5 6 7 8 9 10	tgbtwnconn1	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ∨ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) )
31	20 29 30	mpjaodan	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ∨ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) )