Theorem pwel 5284

Description: Quantitative version of pwexg 5281: 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 5212 and ax-pr 5332 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 5281	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V)
2		elssuni 4870	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)
3	2	sspwd 4556	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵)
4	1, 3	elpwd 4549	1 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3496  𝒫 cpw 4541  ∪ cuni 4840

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-pow 5268

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-in 3945  df-ss 3954  df-pw 4543  df-uni 4841

This theorem is referenced by:  bj-unirel  34346