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	$$⊢A\in \mathrm{V}$$
Assertion	rankpw	$$⊢\mathrm{rank}\left(𝒫A\right)=\mathrm{suc}\mathrm{rank}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		rankpw.1	$$⊢A\in \mathrm{V}$$
2		unir1	$$⊢\bigcup R_1\left[\mathrm{On}\right]=\mathrm{V}$$
3	1 2	eleqtrri	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]$$
4		rankpwi	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]\to \mathrm{rank}\left(𝒫A\right)=\mathrm{suc}\mathrm{rank}\left(A\right)$$
5	3 4	ax-mp	$$⊢\mathrm{rank}\left(𝒫A\right)=\mathrm{suc}\mathrm{rank}\left(A\right)$$