Theorem tgacne

Description: Non-collinearity of triangle A B C implies distinct vertices A and C . (Contributed by JAG, 12-Aug-2026)

	Ref	Expression
Hypotheses	tgacne.p	$$⊢P={\mathrm{Base}}_G$$
	tgacne.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tgacne.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	tgacne.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgacne.a	$$⊢\phi \to A\in P$$
	tgacne.b	$$⊢\phi \to B\in P$$
	tgacne.c	$$⊢\phi \to C\in P$$
	tgacne.ncol	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
Assertion	tgacne	$$⊢\phi \to A\ne C$$

Proof

Step	Hyp	Ref	Expression
1		tgacne.p	$$⊢P={\mathrm{Base}}_G$$
2		tgacne.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
3		tgacne.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
4		tgacne.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		tgacne.a	$$⊢\phi \to A\in P$$
6		tgacne.b	$$⊢\phi \to B\in P$$
7		tgacne.c	$$⊢\phi \to C\in P$$
8		tgacne.ncol	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
9	1 3 2 4 5 6 7 8	ncolrot1	$$⊢\phi \to \neg \left(A\in \left(BLC\right)\vee B=C\right)$$
10	1 3 2 4 6 7 5 9	ncolcom	$$⊢\phi \to \neg \left(A\in \left(CLB\right)\vee C=B\right)$$
11	1 2 3 4 5 7 6 10	ncolne1	$$⊢\phi \to A\ne C$$