Theorem cgranbtwn

Description: Null angle implies betweenness. (Contributed by SS, 4-Jun-2026)

	Ref	Expression
Hypotheses	cgranbtwn.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	cgranbtwn.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	cgranbtwn.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	cgranbtwn.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	cgranbtwn.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	cgranbtwn.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	cgranbtwn.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	cgranbtwn.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
	cgranbtwn.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
	cgranbtwn.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
	cgranbtwn.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐶 ) )
Assertion	cgranbtwn	⊢ ( 𝜑 → ( 𝐷 ∈ ( 𝐸 𝐼 𝐹 ) ∨ 𝐹 ∈ ( 𝐸 𝐼 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cgranbtwn.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		cgranbtwn.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		cgranbtwn.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
4		cgranbtwn.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
5		cgranbtwn.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
6		cgranbtwn.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
7		cgranbtwn.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
8		cgranbtwn.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
9		cgranbtwn.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
10		cgranbtwn.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
11		cgranbtwn.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐶 ) )
12		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
13		eqid	⊢ ( hlG ‘ 𝐺 ) = ( hlG ‘ 𝐺 )
14	1 2 13 3 4 5 6 7 8 9 10	cgrane2	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
15	1 2 13 3 4 5 6 7 8 9 10	cgrane1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
16	1 2 13 5 6 4 3 4 11 14 15	btwnhl1	⊢ ( 𝜑 → 𝐴 ( ( hlG ‘ 𝐺 ) ‘ 𝐵 ) 𝐶 )
17	1 2 12 3 4 5 6 7 8 9 10 13 16	cgrahl	⊢ ( 𝜑 → 𝐷 ( ( hlG ‘ 𝐺 ) ‘ 𝐸 ) 𝐹 )
18	1 2 13 7 9 8 3	ishlg	⊢ ( 𝜑 → ( 𝐷 ( ( hlG ‘ 𝐺 ) ‘ 𝐸 ) 𝐹 ↔ ( 𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ ( 𝐷 ∈ ( 𝐸 𝐼 𝐹 ) ∨ 𝐹 ∈ ( 𝐸 𝐼 𝐷 ) ) ) ) )
19	17 18	mpbid	⊢ ( 𝜑 → ( 𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ ( 𝐷 ∈ ( 𝐸 𝐼 𝐹 ) ∨ 𝐹 ∈ ( 𝐸 𝐼 𝐷 ) ) ) )
20	19	simp3d	⊢ ( 𝜑 → ( 𝐷 ∈ ( 𝐸 𝐼 𝐹 ) ∨ 𝐹 ∈ ( 𝐸 𝐼 𝐷 ) ) )