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 ) ) ) |