Theorem letrii

Description: Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999)

	Ref	Expression
Hypotheses	lt.1	⊢ 𝐴 ∈ ℝ
	lt.2	⊢ 𝐵 ∈ ℝ
Assertion	letrii	⊢ ( 𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lt.1	⊢ 𝐴 ∈ ℝ
2		lt.2	⊢ 𝐵 ∈ ℝ
3	2 1	ltnlei	⊢ ( 𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵 )
4	2 1	ltlei	⊢ ( 𝐵 < 𝐴 → 𝐵 ≤ 𝐴 )
5	3 4	sylbir	⊢ ( ¬ 𝐴 ≤ 𝐵 → 𝐵 ≤ 𝐴 )
6	5	orri	⊢ ( 𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴 )