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Mirrors > Home > MPE Home > Th. List > pwunss | Structured version Visualization version GIF version |
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 5289, ax-nul 5296, ax-pr 5417 and shorten proof. (Revised by BJ, 13-Apr-2024.) |
Ref | Expression |
---|---|
pwunss | ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4164 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | 1 | sspwi 4606 | . 2 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) |
3 | ssun2 4165 | . . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
4 | 3 | sspwi 4606 | . 2 ⊢ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵) |
5 | 2, 4 | unssi 4177 | 1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3938 ⊆ wss 3940 𝒫 cpw 4594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-un 3945 df-in 3947 df-ss 3957 df-pw 4596 |
This theorem is referenced by: pwun 5562 |
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