Description: A relationship between the rank function and the cumulative hierarchy of sets function R1 . Proposition 9.15(2) of TakeutiZaring p. 79. (Contributed by NM, 6-Oct-2003) (Proof shortened by Mario Carneiro, 17-Nov-2014)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | rankid.1 | |- A e. _V |
|
| Assertion | rankr1 | |- ( B = ( rank ` A ) <-> ( -. A e. ( R1 ` B ) /\ A e. ( R1 ` suc B ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rankid.1 | |- A e. _V |
|
| 2 | rankr1g | |- ( A e. _V -> ( B = ( rank ` A ) <-> ( -. A e. ( R1 ` B ) /\ A e. ( R1 ` suc B ) ) ) ) |
|
| 3 | 1 2 | ax-mp | |- ( B = ( rank ` A ) <-> ( -. A e. ( R1 ` B ) /\ A e. ( R1 ` suc B ) ) ) |