Theorem ranksn

Description: The rank of a singleton. Theorem 15.17(v) of Monk1 p. 112. (Contributed by NM, 28-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	ranksn.1	|- A e. _V
Assertion	ranksn	|- ( rank ` { A } ) = suc ( rank ` A )

Proof

Step	Hyp	Ref	Expression
1		ranksn.1	|- A e. _V
2		unir1	|- U. ( R1 " On ) = _V
3	1 2	eleqtrri	|- A e. U. ( R1 " On )
4		ranksnb	|- ( A e. U. ( R1 " On ) -> ( rank ` { A } ) = suc ( rank ` A ) )
5	3 4	ax-mp	|- ( rank ` { A } ) = suc ( rank ` A )