Description: Any set is a subset of the hierarchy of its rank. (Contributed by Mario Carneiro, 3-Jun-2013) (Revised by Mario Carneiro, 17-Nov-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | r1rankidb | |- ( A e. U. ( R1 " On ) -> A C_ ( R1 ` ( rank ` A ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid | |- ( rank ` A ) C_ ( rank ` A ) |
|
2 | rankdmr1 | |- ( rank ` A ) e. dom R1 |
|
3 | rankr1bg | |- ( ( A e. U. ( R1 " On ) /\ ( rank ` A ) e. dom R1 ) -> ( A C_ ( R1 ` ( rank ` A ) ) <-> ( rank ` A ) C_ ( rank ` A ) ) ) |
|
4 | 2 3 | mpan2 | |- ( A e. U. ( R1 " On ) -> ( A C_ ( R1 ` ( rank ` A ) ) <-> ( rank ` A ) C_ ( rank ` A ) ) ) |
5 | 1 4 | mpbiri | |- ( A e. U. ( R1 " On ) -> A C_ ( R1 ` ( rank ` A ) ) ) |