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Metamath Proof Explorer

Theorem elpwunicl

Description: Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020) (Proof shortened by BJ, 6-Apr-2024)

		Ref	Expression
	Hypothesis	elpwunicl.1	⊢ ( 𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵 )
	Assertion	elpwunicl	⊢ ( 𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elpwunicl.1	⊢ ( 𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵 )
2		elpwpwel	⊢ ( 𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵 )
3	1 2	sylib	⊢ ( 𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵 )