Metamath Proof Explorer


Theorem rankr1

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 ) ) )

Proof

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 ) ) )