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 |