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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 7751 | . 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ V) | |
2 | elpwi 4610 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝑋 → 𝐴 ⊆ 𝑋) | |
3 | elpwi 4610 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝑋 → 𝐵 ⊆ 𝑋) | |
4 | unss 4184 | . . . 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 4609 | 1 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 Vcvv 3471 ∪ cun 3945 ⊆ wss 3947 𝒫 cpw 4603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4909 |
This theorem is referenced by: naddunif 8714 fiin 9446 fpwipodrs 18532 pwmnd 18889 cutlt 27865 clsk1indlem3 43473 isotone1 43478 |
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