Theorem rankval3

Description: The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of TakeutiZaring p. 79. (Contributed by NM, 11-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankval3.1	|- A e. _V
Assertion	rankval3	|- ( rank ` A ) = |^| { x e. On | A. y e. A ( rank ` y ) e. x }

Proof

Step	Hyp	Ref	Expression
1		rankval3.1	|- A e. _V
2		unir1	|- U. ( R1 " On ) = _V
3	1 2	eleqtrri	|- A e. U. ( R1 " On )
4		rankval3b	|- ( A e. U. ( R1 " On ) -> ( rank ` A ) = |^| { x e. On | A. y e. A ( rank ` y ) e. x } )
5	3 4	ax-mp	|- ( rank ` A ) = |^| { x e. On | A. y e. A ( rank ` y ) e. x }