Theorem rankr1g

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) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	rankr1g	|- ( A e. V -> ( B = ( rank ` A ) <-> ( -. A e. ( R1 ` B ) /\ A e. ( R1 ` suc B ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		elex	|- ( A e. V -> A e. _V )
2		unir1	|- U. ( R1 " On ) = _V
3	1 2	eleqtrrdi	|- ( A e. V -> A e. U. ( R1 " On ) )
4		rankr1c	|- ( A e. U. ( R1 " On ) -> ( B = ( rank ` A ) <-> ( -. A e. ( R1 ` B ) /\ A e. ( R1 ` suc B ) ) ) )
5	3 4	syl	|- ( A e. V -> ( B = ( rank ` A ) <-> ( -. A e. ( R1 ` B ) /\ A e. ( R1 ` suc B ) ) ) )