Theorem pwunss 4570

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 5243, ax-nul 5253, ax-pr 5387 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 4128	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2	1	sspwi 4564	. 2 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
3		ssun2 4129	. . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4	3	sspwi 4564	. 2 ⊢ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)
5	2, 4	unssi 4141	1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∪ cun 3900   ⊆ wss 3902  𝒫 cpw 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3907  df-ss 3919  df-pw 4554

This theorem is referenced by:  pwun  5536