Theorem nn0expcli

Description: Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015)

	Ref	Expression
Hypotheses	nn0expcli.1	⊢ 𝐴 ∈ ℕ0
	nn0expcli.2	⊢ 𝑁 ∈ ℕ0
Assertion	nn0expcli	⊢ ( 𝐴 ↑ 𝑁 ) ∈ ℕ0

Proof

Step	Hyp	Ref	Expression
1		nn0expcli.1	⊢ 𝐴 ∈ ℕ0
2		nn0expcli.2	⊢ 𝑁 ∈ ℕ0
3		nn0expcl	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℕ0 )
4	1 2 3	mp2an	⊢ ( 𝐴 ↑ 𝑁 ) ∈ ℕ0