Theorem rankpw

Description: The rank of the powerset is the successor of the rank. Part of Exercise 30 of Enderton p. 207. (Contributed by NM, 22-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankpw.1	⊢ 𝐴 ∈ V
Assertion	rankpw	⊢ ( rank ‘ 𝒫 𝐴 ) = suc ( rank ‘ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		rankpw.1	⊢ 𝐴 ∈ V
2		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
3	1 2	eleqtrri	⊢ 𝐴 ∈ ∪ ( 𝑅1 “ On )
4		rankpwi	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ 𝒫 𝐴 ) = suc ( rank ‘ 𝐴 ) )
5	3 4	ax-mp	⊢ ( rank ‘ 𝒫 𝐴 ) = suc ( rank ‘ 𝐴 )