Theorem lttrid

Description: Ordering on reals satisfies strict trichotomy. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltd.2	$$⊢\phi \to B\in ℝ$$
Assertion	lttrid	$$⊢\phi \to \left(A<B\leftrightarrow \neg \left(A=B\vee B<A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		axlttri	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\leftrightarrow \neg \left(A=B\vee B<A\right)\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A<B\leftrightarrow \neg \left(A=B\vee B<A\right)\right)$$