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 5259, ax-nul 5271, ax-pr 5406 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 4131	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2	1	sspwi 4576	. 2 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
3		ssun2 4132	. . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4	3	sspwi 4576	. 2 ⊢ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)
5	2, 4	unssi 4144	1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

This proof depends on syntax axioms:   ∪ cun 3904   ⊆ wss 3906  𝒫 cpw 4564

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-ss 3923  df-pw 4566

This theorem is used by:  pwun  5556