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Metamath Proof Explorer

Theorem rankpw

Description: The rank of a power set. Part of Exercise 30 of Enderton p. 207. (Contributed by NM, 22-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Hypothesis rankpw.1 ⊢ 𝐴 ∈ V
Assertion rankpw ( rank ‘ 𝒫 𝐴 ) = suc ( rank ‘ 𝐴 )

Proof

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