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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 5296, ax-nul 5306, ax-pr 5432 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 4178 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
| 2 | 1 | sspwi 4612 | . 2 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) |
| 3 | ssun2 4179 | . . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
| 4 | 3 | sspwi 4612 | . 2 ⊢ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵) |
| 5 | 2, 4 | unssi 4191 | 1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ∪ cun 3949 ⊆ wss 3951 𝒫 cpw 4600 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-pw 4602 |
| This theorem is referenced by: pwun 5576 |
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