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Mirrors > Home > MPE Home > Th. List > pwuncl | Structured version Visualization version GIF version |
Description: Power classes are closed under union. (Contributed by AV, 27-Feb-2024.) |
Ref | Expression |
---|---|
pwuncl | ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unexg 7732 | . 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ V) | |
2 | elpwi 4604 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝑋 → 𝐴 ⊆ 𝑋) | |
3 | elpwi 4604 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝑋 → 𝐵 ⊆ 𝑋) | |
4 | unss 4179 | . . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋) | |
5 | 4 | biimpi 215 | . . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋) |
6 | 2, 3, 5 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋) |
7 | 1, 6 | elpwd 4603 | 1 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 Vcvv 3468 ∪ cun 3941 ⊆ wss 3943 𝒫 cpw 4597 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-pw 4599 df-sn 4624 df-pr 4626 df-uni 4903 |
This theorem is referenced by: naddunif 8691 fiin 9416 fpwipodrs 18503 pwmnd 18860 cutlt 27803 clsk1indlem3 43351 isotone1 43356 |
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