Theorem pwunss 4547

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 5218, ax-nul 5228, ax-pr 5362 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 4107	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2	1	sspwi 4541	. 2 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
3		ssun2 4108	. . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4	3	sspwi 4541	. 2 ⊢ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)
5	2, 4	unssi 4120	1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∪ cun 3881   ⊆ wss 3883  𝒫 cpw 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-un 3888  df-ss 3900  df-pw 4531

This theorem is referenced by:  pwun  5511