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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 )

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 )