Theorem rankval

Description: Value of the rank function. Definition 9.14 of TakeutiZaring p. 79 (proved as a theorem from our definition). (Contributed by NM, 24-Sep-2003) (Revised by Mario Carneiro, 10-Sep-2013)

	Ref	Expression
Hypothesis	rankval.1	$$⊢A\in \mathrm{V}$$
Assertion	rankval	$$⊢\mathrm{rank}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|A\in R_1\left(\mathrm{suc}x\right)\right\}$$

Proof

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