Theorem pwpwuni 39225

Description: Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
pwpwuni	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))

Proof of Theorem pwpwuni

Step	Hyp	Ref	Expression
1		elpwg 4166	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵))
2		sspwuni 4611	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
4		uniexg 6955	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
5		elpwg 4166	. . . 4 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
7	6	bicomd 213	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))
8	1, 3, 7	3bitrd 294	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  psmeasurelem  40687