Theorem rankun

Description: The rank of the union of two sets. Theorem 15.17(iii) of Monk1 p. 112. (Contributed by NM, 26-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)

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

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		rankunb	$$⊢\left(A\in \bigcup R_1\left[\mathrm{On}\right]\wedge B\in \bigcup R_1\left[\mathrm{On}\right]\right)\to \mathrm{rank}\left(A\cup B\right)=\mathrm{rank}\left(A\right)\cup \mathrm{rank}\left(B\right)$$
7	4 5 6	mp2an	$$⊢\mathrm{rank}\left(A\cup B\right)=\mathrm{rank}\left(A\right)\cup \mathrm{rank}\left(B\right)$$