Metamath Proof Explorer


Theorem nnm1nn0

Description: A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007) (Revised by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion nnm1nn0
|- ( N e. NN -> ( N - 1 ) e. NN0 )

Proof

Step Hyp Ref Expression
1 nn1m1nn
 |-  ( N e. NN -> ( N = 1 \/ ( N - 1 ) e. NN ) )
2 oveq1
 |-  ( N = 1 -> ( N - 1 ) = ( 1 - 1 ) )
3 1m1e0
 |-  ( 1 - 1 ) = 0
4 2 3 syl6eq
 |-  ( N = 1 -> ( N - 1 ) = 0 )
5 4 orim1i
 |-  ( ( N = 1 \/ ( N - 1 ) e. NN ) -> ( ( N - 1 ) = 0 \/ ( N - 1 ) e. NN ) )
6 1 5 syl
 |-  ( N e. NN -> ( ( N - 1 ) = 0 \/ ( N - 1 ) e. NN ) )
7 6 orcomd
 |-  ( N e. NN -> ( ( N - 1 ) e. NN \/ ( N - 1 ) = 0 ) )
8 elnn0
 |-  ( ( N - 1 ) e. NN0 <-> ( ( N - 1 ) e. NN \/ ( N - 1 ) = 0 ) )
9 7 8 sylibr
 |-  ( N e. NN -> ( N - 1 ) e. NN0 )