Theorem lttrid

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

	Ref	Expression
Hypotheses	ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	lttrid	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		axlttri	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) )