Theorem elpwunicl 28754

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

Hypotheses

Ref	Expression
elpwunicl.1	⊢ (𝜑 → 𝐵 ∈ 𝑉)
elpwunicl.2	⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵)

Assertion

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

Proof of Theorem elpwunicl

Step	Hyp	Ref	Expression
1		elpwunicl.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵)
2		elpwg 4116	. . . . 5 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵))
4	1, 3	mpbid 221	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝐵)
5		pwuniss 28753	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ 𝐵)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → ∪ 𝐴 ⊆ 𝐵)
7		uniexg 6853	. . 3 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 → ∪ 𝐴 ∈ V)
8		elpwg 4116	. . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
9	1, 7, 8	3syl 18	. 2 ⊢ (𝜑 → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
10	6, 9	mpbird 246	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373

This theorem is referenced by:  ldgenpisyslem1  29553