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 ∈ V
Assertion	rankuni2	⊢ ( rank ‘ ∪ 𝐴 ) = ∪ 𝑥 ∈ 𝐴 ( rank ‘ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ranksn.1	⊢ 𝐴 ∈ V
2		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
3	1 2	eleqtrri	⊢ 𝐴 ∈ ∪ ( 𝑅1 “ On )
4		rankuni2b	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ ∪ 𝐴 ) = ∪ 𝑥 ∈ 𝐴 ( rank ‘ 𝑥 ) )
5	3 4	ax-mp	⊢ ( rank ‘ ∪ 𝐴 ) = ∪ 𝑥 ∈ 𝐴 ( rank ‘ 𝑥 )