Theorem tgaasncol

Description: Slide a non-collinearity along the line it is off: if X is off line Y Z and W is a point of that line other than Z , then X is off line Z W too. Lines Z Y and Z W coincide ( tglineeltr ), which is what the exterior-angle hypothesis of tgaasexc needs at the moving point. (Contributed by JAG, 12-Aug-2026)

	Ref	Expression
Hypotheses	tgaasncol.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tgaasncol.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tgaasncol.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	tgaasncol.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgaasncol.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	tgaasncol.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
	tgaasncol.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
	tgaasncol.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑃 )
	tgaasncol.1	⊢ ( 𝜑 → ¬ ( 𝑋 ∈ ( 𝑌 𝐿 𝑍 ) ∨ 𝑌 = 𝑍 ) )
	tgaasncol.2	⊢ ( 𝜑 → ( 𝑊 ∈ ( 𝑍 𝐿 𝑌 ) ∨ 𝑍 = 𝑌 ) )
	tgaasncol.3	⊢ ( 𝜑 → 𝑊 ≠ 𝑍 )
Assertion	tgaasncol	⊢ ( 𝜑 → ¬ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		tgaasncol.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tgaasncol.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		tgaasncol.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
4		tgaasncol.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgaasncol.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
6		tgaasncol.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
7		tgaasncol.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
8		tgaasncol.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑃 )
9		tgaasncol.1	⊢ ( 𝜑 → ¬ ( 𝑋 ∈ ( 𝑌 𝐿 𝑍 ) ∨ 𝑌 = 𝑍 ) )
10		tgaasncol.2	⊢ ( 𝜑 → ( 𝑊 ∈ ( 𝑍 𝐿 𝑌 ) ∨ 𝑍 = 𝑌 ) )
11		tgaasncol.3	⊢ ( 𝜑 → 𝑊 ≠ 𝑍 )
12	1 3 2 4 6 7 5 9	ncolcom	⊢ ( 𝜑 → ¬ ( 𝑋 ∈ ( 𝑍 𝐿 𝑌 ) ∨ 𝑍 = 𝑌 ) )
13	4	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝐺 ∈ TarskiG )
14	7	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑍 ∈ 𝑃 )
15	6	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑌 ∈ 𝑃 )
16	1 3 2 4 6 7 5 9	ncolrot1	⊢ ( 𝜑 → ¬ ( 𝑌 ∈ ( 𝑍 𝐿 𝑋 ) ∨ 𝑍 = 𝑋 ) )
17	1 2 3 4 6 7 5 16	ncolne1	⊢ ( 𝜑 → 𝑌 ≠ 𝑍 )
18	17	necomd	⊢ ( 𝜑 → 𝑍 ≠ 𝑌 )
19	18	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑍 ≠ 𝑌 )
20	8	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑊 ∈ 𝑃 )
21	11	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑊 ≠ 𝑍 )
22	18	neneqd	⊢ ( 𝜑 → ¬ 𝑍 = 𝑌 )
23	10 22	olcnd	⊢ ( 𝜑 → 𝑊 ∈ ( 𝑍 𝐿 𝑌 ) )
24	23	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑊 ∈ ( 𝑍 𝐿 𝑌 ) )
25	5	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑋 ∈ 𝑃 )
26		simpr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) )
27	21	necomd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑍 ≠ 𝑊 )
28	27	neneqd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → ¬ 𝑍 = 𝑊 )
29	26 28	olcnd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) )
30	1 2 3 13 14 15 19 20 21 24 25 29	tglineeltr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → 𝑋 ∈ ( 𝑍 𝐿 𝑌 ) )
31	30	orcd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) ) → ( 𝑋 ∈ ( 𝑍 𝐿 𝑌 ) ∨ 𝑍 = 𝑌 ) )
32	12 31	mtand	⊢ ( 𝜑 → ¬ ( 𝑋 ∈ ( 𝑍 𝐿 𝑊 ) ∨ 𝑍 = 𝑊 ) )