Metamath Proof Explorer

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 e. NN0 -> ( N + 1 ) e. NN )

Proof

Step Hyp Ref Expression
1 1nn 
 |-  1 e. NN
2 nn0nnaddcl 
 |-  ( ( N e. NN0 /\ 1 e. NN ) -> ( N + 1 ) e. NN )
3 1 2 mpan2 
 |-  ( N e. NN0 -> ( N + 1 ) e. NN )