Theorem tgbcne

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

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

Proof

Step	Hyp	Ref	Expression
1		tgbcne.p	$$⊢P={\mathrm{Base}}_G$$
2		tgbcne.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
3		tgbcne.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
4		tgbcne.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		tgbcne.a	$$⊢\phi \to A\in P$$
6		tgbcne.b	$$⊢\phi \to B\in P$$
7		tgbcne.c	$$⊢\phi \to C\in P$$
8		tgbcne.ncol	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
9	1 3 2 4 5 6 7 8	ncolrot2	$$⊢\phi \to \neg \left(B\in \left(CLA\right)\vee C=A\right)$$
10	1 2 3 4 6 7 5 9	ncolne1	$$⊢\phi \to B\ne C$$