Theorem rankval3

Description: The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of TakeutiZaring p. 79. (Contributed by NM, 11-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankval3.1	$$⊢A\in \mathrm{V}$$
Assertion	rankval3	$$⊢\mathrm{rank}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|\forall y\in A\mathrm{rank}\left(y\right)\in x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		rankval3.1	$$⊢A\in \mathrm{V}$$
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		rankval3b	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]\to \mathrm{rank}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|\forall y\in A\mathrm{rank}\left(y\right)\in x\right\}$$
5	3 4	ax-mp	$$⊢\mathrm{rank}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|\forall y\in A\mathrm{rank}\left(y\right)\in x\right\}$$