Theorem sotrine

Description: Trichotomy law for strict orderings. (Contributed by Scott Fenton, 8-Dec-2021)

	Ref	Expression
Assertion	sotrine	$$⊢\left(R\mathrm{Or}A\wedge \left(B\in A\wedge C\in A\right)\right)\to \left(B\ne C\leftrightarrow \left(BRC\vee CRB\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sotrieq	$$⊢\left(R\mathrm{Or}A\wedge \left(B\in A\wedge C\in A\right)\right)\to \left(B=C\leftrightarrow \neg \left(BRC\vee CRB\right)\right)$$
2	1	bicomd	$$⊢\left(R\mathrm{Or}A\wedge \left(B\in A\wedge C\in A\right)\right)\to \left(\neg \left(BRC\vee CRB\right)\leftrightarrow B=C\right)$$
3	2	necon1abid	$$⊢\left(R\mathrm{Or}A\wedge \left(B\in A\wedge C\in A\right)\right)\to \left(B\ne C\leftrightarrow \left(BRC\vee CRB\right)\right)$$