Theorem lttri4d

Description: Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltd.2	$$⊢\phi \to B\in ℝ$$
Assertion	lttri4d	$$⊢\phi \to \left(A<B\vee A=B\vee B<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		lttri4	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\vee A=B\vee B<A\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A<B\vee A=B\vee B<A\right)$$