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 ∈ V
	rankun.2	⊢ 𝐵 ∈ V
Assertion	rankun	⊢ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ranksn.1	⊢ 𝐴 ∈ V
2		rankun.2	⊢ 𝐵 ∈ V
3		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
4	1 3	eleqtrri	⊢ 𝐴 ∈ ∪ ( 𝑅1 “ On )
5	2 3	eleqtrri	⊢ 𝐵 ∈ ∪ ( 𝑅1 “ On )
6		rankunb	⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) )
7	4 5 6	mp2an	⊢ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) )