# Metamath Proof Explorer

## Theorem elnn0nn

Description: The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004) (Proof shortened by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion elnn0nn
`|- ( N e. NN0 <-> ( N e. CC /\ ( N + 1 ) e. NN ) )`

### Proof

Step Hyp Ref Expression
1 nn0cn
` |-  ( N e. NN0 -> N e. CC )`
2 nn0p1nn
` |-  ( N e. NN0 -> ( N + 1 ) e. NN )`
3 1 2 jca
` |-  ( N e. NN0 -> ( N e. CC /\ ( N + 1 ) e. NN ) )`
4 simpl
` |-  ( ( N e. CC /\ ( N + 1 ) e. NN ) -> N e. CC )`
5 ax-1cn
` |-  1 e. CC`
6 pncan
` |-  ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - 1 ) = N )`
7 4 5 6 sylancl
` |-  ( ( N e. CC /\ ( N + 1 ) e. NN ) -> ( ( N + 1 ) - 1 ) = N )`
8 nnm1nn0
` |-  ( ( N + 1 ) e. NN -> ( ( N + 1 ) - 1 ) e. NN0 )`
9 8 adantl
` |-  ( ( N e. CC /\ ( N + 1 ) e. NN ) -> ( ( N + 1 ) - 1 ) e. NN0 )`
10 7 9 eqeltrrd
` |-  ( ( N e. CC /\ ( N + 1 ) e. NN ) -> N e. NN0 )`
11 3 10 impbii
` |-  ( N e. NN0 <-> ( N e. CC /\ ( N + 1 ) e. NN ) )`