Description: An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | uzct.1 | |- Z = ( ZZ>= ` N ) |
|
| Assertion | uzct | |- Z ~<_ _om |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uzct.1 | |- Z = ( ZZ>= ` N ) |
|
| 2 | uzssz | |- ( ZZ>= ` N ) C_ ZZ |
|
| 3 | 1 2 | eqsstri | |- Z C_ ZZ |
| 4 | zex | |- ZZ e. _V |
|
| 5 | ssdomg | |- ( ZZ e. _V -> ( Z C_ ZZ -> Z ~<_ ZZ ) ) |
|
| 6 | 4 5 | ax-mp | |- ( Z C_ ZZ -> Z ~<_ ZZ ) |
| 7 | 3 6 | ax-mp | |- Z ~<_ ZZ |
| 8 | zct | |- ZZ ~<_ _om |
|
| 9 | domtr | |- ( ( Z ~<_ ZZ /\ ZZ ~<_ _om ) -> Z ~<_ _om ) |
|
| 10 | 7 8 9 | mp2an | |- Z ~<_ _om |