Theorem rankpwg

Description: The rank of the powerset is the successor of the rank. Closed form of rankpw . (Contributed by Scott Fenton, 16-Jul-2015)

	Ref	Expression
Assertion	rankpwg	$$⊢A\in V\to \mathrm{rank}\left(𝒫A\right)=\mathrm{suc}\mathrm{rank}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		pweq	$$⊢x=A\to 𝒫x=𝒫A$$
2	1	fveq2d	$$⊢x=A\to \mathrm{rank}\left(𝒫x\right)=\mathrm{rank}\left(𝒫A\right)$$
3		fveq2	$$⊢x=A\to \mathrm{rank}\left(x\right)=\mathrm{rank}\left(A\right)$$
4		suceq	$$⊢\mathrm{rank}\left(x\right)=\mathrm{rank}\left(A\right)\to \mathrm{suc}\mathrm{rank}\left(x\right)=\mathrm{suc}\mathrm{rank}\left(A\right)$$
5	3 4	syl	$$⊢x=A\to \mathrm{suc}\mathrm{rank}\left(x\right)=\mathrm{suc}\mathrm{rank}\left(A\right)$$
6	2 5	eqeq12d	$$⊢x=A\to \left(\mathrm{rank}\left(𝒫x\right)=\mathrm{suc}\mathrm{rank}\left(x\right)\leftrightarrow \mathrm{rank}\left(𝒫A\right)=\mathrm{suc}\mathrm{rank}\left(A\right)\right)$$
7		vex	$$⊢x\in \mathrm{V}$$
8	7	rankpw	$$⊢\mathrm{rank}\left(𝒫x\right)=\mathrm{suc}\mathrm{rank}\left(x\right)$$
9	6 8	vtoclg	$$⊢A\in V\to \mathrm{rank}\left(𝒫A\right)=\mathrm{suc}\mathrm{rank}\left(A\right)$$