Theorem elpwunicl 32502

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 7769	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)
3	1, 2	sylib 218	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2107  𝒫 cpw 4580  ∪ cuni 4887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-pow 5345  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rab 3420  df-v 3465  df-in 3938  df-ss 3948  df-pw 4582  df-uni 4888

This theorem is referenced by:  ldgenpisyslem1  34123