Theorem rankop

Description: The rank of an ordered pair. Part of Exercise 4 of Kunen p. 107. (Contributed by NM, 13-Sep-2006) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypotheses	ranksn.1	$$⊢A\in \mathrm{V}$$
	rankun.2	$$⊢B\in \mathrm{V}$$
Assertion	rankop	$$⊢\mathrm{rank}\left(⟨A,B⟩\right)=\mathrm{suc}\mathrm{suc}\left(\mathrm{rank}\left(A\right)\cup \mathrm{rank}\left(B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ranksn.1	$$⊢A\in \mathrm{V}$$
2		rankun.2	$$⊢B\in \mathrm{V}$$
3		unir1	$$⊢\bigcup R_1\left[\mathrm{On}\right]=\mathrm{V}$$
4	1 3	eleqtrri	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]$$
5	2 3	eleqtrri	$$⊢B\in \bigcup R_1\left[\mathrm{On}\right]$$
6		rankopb	$$⊢\left(A\in \bigcup R_1\left[\mathrm{On}\right]\wedge B\in \bigcup R_1\left[\mathrm{On}\right]\right)\to \mathrm{rank}\left(⟨A,B⟩\right)=\mathrm{suc}\mathrm{suc}\left(\mathrm{rank}\left(A\right)\cup \mathrm{rank}\left(B\right)\right)$$
7	4 5 6	mp2an	$$⊢\mathrm{rank}\left(⟨A,B⟩\right)=\mathrm{suc}\mathrm{suc}\left(\mathrm{rank}\left(A\right)\cup \mathrm{rank}\left(B\right)\right)$$