Theorem bj-unirel 37093

Description: Quantitative version of uniexr 7696: if the union of a class is an element of a class, then that class is an element of the double powerclass of the union of this class. (Contributed by BJ, 6-Apr-2024.)

Assertion

Ref	Expression
bj-unirel	⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)

Proof of Theorem bj-unirel

Step	Hyp	Ref	Expression
1		pwuni 4894	. . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		pwel 5317	. . 3 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)
3		bj-sselpwuni 37092	. . 3 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉)
4	1, 2, 3	sylancr 587	. 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉)
5		unipw 5389	. . 3 ⊢ ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 ∪ 𝑉
6	5	pweqi 4563	. 2 ⊢ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 𝒫 ∪ 𝑉
7	4, 6	eleqtrdi 2841	1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)

Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-pow 5301  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-un 3902  df-in 3904  df-ss 3914  df-pw 4549  df-sn 4574  df-pr 4576  df-uni 4857

This theorem is referenced by: (None)