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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 5203, ax-nul 5210, ax-pr 5330 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 4148 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | 1 | sspwi 4553 | . 2 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) |
3 | ssun2 4149 | . . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
4 | 3 | sspwi 4553 | . 2 ⊢ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵) |
5 | 2, 4 | unssi 4161 | 1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
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 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3941 df-in 3943 df-ss 3952 df-pw 4541 |
This theorem is referenced by: pwundifOLD 5457 pwun 5458 |
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