Theorem nn0p1nn

Description: A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn . (Contributed by Raph Levien, 30-Jun-2006) (Revised by Mario Carneiro, 16-May-2014)

	Ref	Expression
Assertion	nn0p1nn	$$⊢N\in {\mathbb{N}}_0\to N+1\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		1nn	$$⊢1\in \mathbb{N}$$
2		nn0nnaddcl	$$⊢\left(N\in {\mathbb{N}}_0\wedge 1\in \mathbb{N}\right)\to N+1\in \mathbb{N}$$
3	1 2	mpan2	$$⊢N\in {\mathbb{N}}_0\to N+1\in \mathbb{N}$$