Theorem rankid

Description: Identity law for the rank function. (Contributed by NM, 3-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankid.1	$$⊢A\in \mathrm{V}$$
Assertion	rankid	$$⊢A\in R_1\left(\mathrm{suc}\mathrm{rank}\left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		rankid.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		rankidb	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]\to A\in R_1\left(\mathrm{suc}\mathrm{rank}\left(A\right)\right)$$
5	3 4	ax-mp	$$⊢A\in R_1\left(\mathrm{suc}\mathrm{rank}\left(A\right)\right)$$