Theorem rankuni2

Description: The rank of a union. Part of Theorem 15.17(iv) of Monk1 p. 112. (Contributed by NM, 30-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	ranksn.1	$$⊢A\in \mathrm{V}$$
Assertion	rankuni2	$$⊢\mathrm{rank}\left(\bigcup A\right)=\bigcup _{x\in A}\mathrm{rank}\left(x\right)$$

Proof

Step	Hyp	Ref	Expression
1		ranksn.1	$$⊢A\in \mathrm{V}$$
2		unir1	$$⊢\bigcup R_1\left[\mathrm{On}\right]=\mathrm{V}$$
3	1 2	eleqtrri	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]$$
4		rankuni2b	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]\to \mathrm{rank}\left(\bigcup A\right)=\bigcup _{x\in A}\mathrm{rank}\left(x\right)$$
5	3 4	ax-mp	$$⊢\mathrm{rank}\left(\bigcup A\right)=\bigcup _{x\in A}\mathrm{rank}\left(x\right)$$