Description: Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004) (Proof shortened by Mario Carneiro, 16-May-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | nn0ltlem1 | |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> M <_ ( N - 1 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z | |- ( M e. NN0 -> M e. ZZ ) |
|
2 | nn0z | |- ( N e. NN0 -> N e. ZZ ) |
|
3 | zltlem1 | |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) ) |
|
4 | 1 2 3 | syl2an | |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> M <_ ( N - 1 ) ) ) |