Theorem uznn0sub

Description: The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005)

	Ref	Expression
Assertion	uznn0sub	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 − 𝑀 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		eluz2	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) )
2		znn0sub	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑁 − 𝑀 ) ∈ ℕ0 ) )
3	2	biimp3a	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) → ( 𝑁 − 𝑀 ) ∈ ℕ0 )
4	1 3	sylbi	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 − 𝑀 ) ∈ ℕ0 )