Theorem ranksng

Description: The rank of a singleton. Closed form of ranksn . (Contributed by Scott Fenton, 15-Jul-2015)

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

Proof

Step	Hyp	Ref	Expression
1		sneq	$$⊢x=A\to \left\{x\right\}=\left\{A\right\}$$
2	1	fveq2d	$$⊢x=A\to \mathrm{rank}\left(\left\{x\right\}\right)=\mathrm{rank}\left(\left\{A\right\}\right)$$
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	$$⊢x=A\to \left(\mathrm{rank}\left(\left\{x\right\}\right)=\mathrm{suc}\mathrm{rank}\left(x\right)\leftrightarrow \mathrm{rank}\left(\left\{A\right\}\right)=\mathrm{suc}\mathrm{rank}\left(A\right)\right)$$
7		vex	$$⊢x\in \mathrm{V}$$
8	7	ranksn	$$⊢\mathrm{rank}\left(\left\{x\right\}\right)=\mathrm{suc}\mathrm{rank}\left(x\right)$$
9	6 8	vtoclg	$$⊢A\in V\to \mathrm{rank}\left(\left\{A\right\}\right)=\mathrm{suc}\mathrm{rank}\left(A\right)$$