Theorem elpwunicl 32629

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 7712	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)
3	1, 2	sylib 218	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2113  𝒫 cpw 4554  ∪ cuni 4863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-pow 5310  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918  df-pw 4556  df-uni 4864

This theorem is referenced by:  ldgenpisyslem1  34320