Theorem nnsubi

Description: Subtraction of positive integers. (Contributed by NM, 19-Aug-2001)

	Ref	Expression
Hypotheses	nnsub.1	⊢ 𝐴 ∈ ℕ
	nnsub.2	⊢ 𝐵 ∈ ℕ
Assertion	nnsubi	⊢ ( 𝐴 < 𝐵 ↔ ( 𝐵 − 𝐴 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		nnsub.1	⊢ 𝐴 ∈ ℕ
2		nnsub.2	⊢ 𝐵 ∈ ℕ
3		nnsub	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 < 𝐵 ↔ ( 𝐵 − 𝐴 ) ∈ ℕ ) )
4	1 2 3	mp2an	⊢ ( 𝐴 < 𝐵 ↔ ( 𝐵 − 𝐴 ) ∈ ℕ )