Theorem bj-unirel 37039

Description: Quantitative version of uniexr 7739: 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 4909	. . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		pwel 5336	. . 3 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)
3		bj-sselpwuni 37038	. . 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 5410	. . 3 ⊢ ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 ∪ 𝑉
6	5	pweqi 4579	. 2 ⊢ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 𝒫 ∪ 𝑉
7	4, 6	eleqtrdi 2838	1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)

Syntax hints:   → wi 4   ∈ wcel 2109   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-pow 5320  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-un 3919  df-in 3921  df-ss 3931  df-pw 4565  df-sn 4590  df-pr 4592  df-uni 4872

This theorem is referenced by: (None)