Theorem hlne1

Description: The half-line relation implies inequality. (Contributed by Thierry Arnoux, 22-Feb-2020)

	Ref	Expression
Hypotheses	ishlg.p	$$⊢P={\mathrm{Base}}_G$$
	ishlg.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	ishlg.k	$$⊢K={hl}_{𝒢}\left(G\right)$$
	ishlg.a	$$⊢\phi \to A\in P$$
	ishlg.b	$$⊢\phi \to B\in P$$
	ishlg.c	$$⊢\phi \to C\in P$$
	ishlg.g	$$⊢\phi \to G\in V$$
	hlcomd.1	$$⊢\phi \to AK\left(C\right)B$$
Assertion	hlne1	$$⊢\phi \to A\ne C$$

Proof

Step	Hyp	Ref	Expression
1		ishlg.p	$$⊢P={\mathrm{Base}}_G$$
2		ishlg.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
3		ishlg.k	$$⊢K={hl}_{𝒢}\left(G\right)$$
4		ishlg.a	$$⊢\phi \to A\in P$$
5		ishlg.b	$$⊢\phi \to B\in P$$
6		ishlg.c	$$⊢\phi \to C\in P$$
7		ishlg.g	$$⊢\phi \to G\in V$$
8		hlcomd.1	$$⊢\phi \to AK\left(C\right)B$$
9	1 2 3 4 5 6 7	ishlg	$$⊢\phi \to \left(AK\left(C\right)B\leftrightarrow \left(A\ne C\wedge B\ne C\wedge \left(A\in \left(CIB\right)\vee B\in \left(CIA\right)\right)\right)\right)$$
10	8 9	mpbid	$$⊢\phi \to \left(A\ne C\wedge B\ne C\wedge \left(A\in \left(CIB\right)\vee B\in \left(CIA\right)\right)\right)$$
11	10	simp1d	$$⊢\phi \to A\ne C$$