Theorem ssrankr1

Description: A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets R1 . Proposition 9.15(3) of TakeutiZaring p. 79. (Contributed by NM, 8-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankid.1	|- A e. _V
Assertion	ssrankr1	|- ( B e. On -> ( B C_ ( rank ` A ) <-> -. A e. ( R1 ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		rankid.1	|- A e. _V
2		unir1	|- U. ( R1 " On ) = _V
3	1 2	eleqtrri	|- A e. U. ( R1 " On )
4		r1fnon	|- R1 Fn On
5		fndm	|- ( R1 Fn On -> dom R1 = On )
6	4 5	ax-mp	|- dom R1 = On
7	6	eleq2i	|- ( B e. dom R1 <-> B e. On )
8	7	biimpri	|- ( B e. On -> B e. dom R1 )
9		rankr1clem	|- ( ( A e. U. ( R1 " On ) /\ B e. dom R1 ) -> ( -. A e. ( R1 ` B ) <-> B C_ ( rank ` A ) ) )
10	3 8 9	sylancr	|- ( B e. On -> ( -. A e. ( R1 ` B ) <-> B C_ ( rank ` A ) ) )
11	10	bicomd	|- ( B e. On -> ( B C_ ( rank ` A ) <-> -. A e. ( R1 ` B ) ) )