Description: A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006) (Revised by Mario Carneiro, 17-Nov-2014)
Ref | Expression | ||
---|---|---|---|
Hypothesis | rankeq0.1 | |- A e. _V |
|
Assertion | rankeq0 | |- ( A = (/) <-> ( rank ` A ) = (/) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankeq0.1 | |- A e. _V |
|
2 | unir1 | |- U. ( R1 " On ) = _V |
|
3 | 1 2 | eleqtrri | |- A e. U. ( R1 " On ) |
4 | rankeq0b | |- ( A e. U. ( R1 " On ) -> ( A = (/) <-> ( rank ` A ) = (/) ) ) |
|
5 | 3 4 | ax-mp | |- ( A = (/) <-> ( rank ` A ) = (/) ) |