Theorem bj-unirel 37746

Description: Quantitative version of uniexr 7768: 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 4913	. . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		pwel 5354	. . 3 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)
3		bj-sselpwuni 37745	. . 3 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉)
4	1, 2, 3	sylancr 599	. 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉)
5		unipw 5433	. . 3 ⊢ ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 ∪ 𝑉
6	5	pweqi 4580	. 2 ⊢ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 𝒫 ∪ 𝑉
7	4, 6	eleqtrdi 2875	1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ⊆ wss 3906  𝒫 cpw 4564  ∪ cuni 4874

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-un 3911  df-in 3913  df-ss 3923  df-pw 4566  df-sn 4592  df-pr 4594  df-uni 4875

This theorem is used by: (None)