Theorem pwel 5356

Description: Quantitative version of pwexg 5353: the powerset of an element of a class is an element of the double powerclass of the union of that class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) Remove use of ax-nul 5281 and ax-pr 5407 and shorten proof. (Revised by BJ, 13-Apr-2024.)

Assertion

Ref	Expression
pwel	⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)

Proof of Theorem pwel

Step	Hyp	Ref	Expression
1		pwexg 5353	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V)
2		elssuni 4918	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)
3	2	sspwd 4593	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵)
4	1, 3	elpwd 4586	1 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3464  𝒫 cpw 4580  ∪ cuni 4888

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 2708  ax-sep 5271  ax-pow 5340

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-ss 3948  df-pw 4582  df-uni 4889

This theorem is referenced by:  bj-unirel  37074