Theorem elpwunicl 32946

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.)

Hypothesis

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

Assertion

Ref	Expression
elpwunicl	⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwunicl

Step	Hyp	Ref	Expression
1		elpwunicl.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵)
2		elpwpwel 7768	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)
3	1, 2	sylib 221	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  𝒫 cpw 4564  ∪ cuni 4874

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-in 3913  df-ss 3923  df-pw 4566  df-uni 4875

This theorem is used by:  ldgenpisyslem1  34594