Metamath Proof Explorer


Theorem onpwsuc

Description: The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004)

Ref Expression
Assertion onpwsuc ( 𝐴 ∈ On → ( 𝒫 𝐴 ∩ On ) = suc 𝐴 )

Proof

Step Hyp Ref Expression
1 eloni ( 𝐴 ∈ On → Ord 𝐴 )
2 ordpwsuc ( Ord 𝐴 → ( 𝒫 𝐴 ∩ On ) = suc 𝐴 )
3 1 2 syl ( 𝐴 ∈ On → ( 𝒫 𝐴 ∩ On ) = suc 𝐴 )