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Metamath Proof Explorer

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 )