Theorem pwunss 4582

Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) Remove use of ax-sep 5258, ax-nul 5268, ax-pr 5402 and shorten proof. (Revised by BJ, 13-Apr-2024.)

Assertion

Ref	Expression
pwunss	⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Proof of Theorem pwunss

Step	Hyp	Ref	Expression
1		ssun1 4139	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2	1	sspwi 4576	. 2 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
3		ssun2 4140	. . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4	3	sspwi 4576	. 2 ⊢ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)
5	2, 4	unssi 4152	1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∪ cun 3911   ⊆ wss 3913  𝒫 cpw 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-pw 4566

This theorem is referenced by:  pwun  5552