Theorem subge0i

Description: Nonnegative subtraction. (Contributed by NM, 13-Aug-2000)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
Assertion	subge0i	⊢ ( 0 ≤ ( 𝐴 − 𝐵 ) ↔ 𝐵 ≤ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		subge0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ ( 𝐴 − 𝐵 ) ↔ 𝐵 ≤ 𝐴 ) )
4	1 2 3	mp2an	⊢ ( 0 ≤ ( 𝐴 − 𝐵 ) ↔ 𝐵 ≤ 𝐴 )