Theorem r1rankid

Description: Any set is a subset of the hierarchy of its rank. (Contributed by NM, 14-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	r1rankid	$$⊢A\in V\to A\subseteq R_1\left(\mathrm{rank}\left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in V\to A\in \mathrm{V}$$
2		unir1	$$⊢\bigcup R_1\left[\mathrm{On}\right]=\mathrm{V}$$
3	1 2	eleqtrrdi	$$⊢A\in V\to A\in \bigcup R_1\left[\mathrm{On}\right]$$
4		r1rankidb	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]\to A\subseteq R_1\left(\mathrm{rank}\left(A\right)\right)$$
5	3 4	syl	$$⊢A\in V\to A\subseteq R_1\left(\mathrm{rank}\left(A\right)\right)$$