Theorem pwunss 5448

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.) (Proof shortened by BJ, 30-Dec-2023.)

Assertion

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

Proof of Theorem pwunss

Step	Hyp	Ref	Expression
1		ssun1 4148	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		sspwb 5334	. . 3 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵))
3	1, 2	mpbi 232	. 2 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
4		ssun2 4149	. . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
5		sspwb 5334	. . 3 ⊢ (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵))
6	4, 5	mpbi 232	. 2 ⊢ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)
7	3, 6	unssi 4161	1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∪ cun 3934   ⊆ wss 3936  𝒫 cpw 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-pw 4541  df-sn 4562  df-pr 4564

This theorem is referenced by:  pwundifOLD  5452  pwun  5453