Description: The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007) (Proof shortened by Mario Carneiro, 13-Feb-2013)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dfn2 | |- NN = ( NN0 \ { 0 } ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-n0 | |- NN0 = ( NN u. { 0 } ) |
|
| 2 | 1 | difeq1i | |- ( NN0 \ { 0 } ) = ( ( NN u. { 0 } ) \ { 0 } ) |
| 3 | difun2 | |- ( ( NN u. { 0 } ) \ { 0 } ) = ( NN \ { 0 } ) |
|
| 4 | 0nnn | |- -. 0 e. NN |
|
| 5 | difsn | |- ( -. 0 e. NN -> ( NN \ { 0 } ) = NN ) |
|
| 6 | 4 5 | ax-mp | |- ( NN \ { 0 } ) = NN |
| 7 | 2 3 6 | 3eqtrri | |- NN = ( NN0 \ { 0 } ) |