Theorem rankelg

Description: The membership relation is inherited by the rank function. Closed form of rankel . (Contributed by Scott Fenton, 16-Jul-2015)

	Ref	Expression
Assertion	rankelg	$$⊢\left(B\in V\wedge A\in B\right)\to \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$$⊢y=B\to \left(A\in y\leftrightarrow A\in B\right)$$
2		fveq2	$$⊢y=B\to \mathrm{rank}\left(y\right)=\mathrm{rank}\left(B\right)$$
3	2	eleq2d	$$⊢y=B\to \left(\mathrm{rank}\left(A\right)\in \mathrm{rank}\left(y\right)\leftrightarrow \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(B\right)\right)$$
4	1 3	imbi12d	$$⊢y=B\to \left(\left(A\in y\to \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(y\right)\right)\leftrightarrow \left(A\in B\to \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(B\right)\right)\right)$$
5		vex	$$⊢y\in \mathrm{V}$$
6	5	rankel	$$⊢A\in y\to \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(y\right)$$
7	4 6	vtoclg	$$⊢B\in V\to \left(A\in B\to \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(B\right)\right)$$
8	7	imp	$$⊢\left(B\in V\wedge A\in B\right)\to \mathrm{rank}\left(A\right)\in \mathrm{rank}\left(B\right)$$