Theorem rankel

Description: The membership relation is inherited by the rank function. Proposition 9.16 of TakeutiZaring p. 79. (Contributed by NM, 4-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankel.1	$$⊢B\in \mathrm{V}$$
Assertion	rankel	$$⊢A\in B\to \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(B\right)$$

Proof

Step	Hyp	Ref	Expression
1		rankel.1	$$⊢B\in \mathrm{V}$$
2		unir1	$$⊢\bigcup R_1\left[\mathrm{On}\right]=\mathrm{V}$$
3	1 2	eleqtrri	$$⊢B\in \bigcup R_1\left[\mathrm{On}\right]$$
4		rankelb	$$⊢B\in \bigcup R_1\left[\mathrm{On}\right]\to \left(A\in B\to \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(B\right)\right)$$
5	3 4	ax-mp	$$⊢A\in B\to \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(B\right)$$