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| 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 7764 | . 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ V) | |
| 2 | elpwi 4606 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝑋 → 𝐴 ⊆ 𝑋) | |
| 3 | elpwi 4606 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝑋 → 𝐵 ⊆ 𝑋) | |
| 4 | unss 4189 | . . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋) | |
| 5 | 4 | biimpi 216 | . . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋) | 
| 6 | 2, 3, 5 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋) | 
| 7 | 1, 6 | elpwd 4605 | 1 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 ⊆ wss 3950 𝒫 cpw 4599 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-pw 4601 df-sn 4626 df-pr 4628 df-uni 4907 | 
| This theorem is referenced by: naddunif 8732 fiin 9463 fpwipodrs 18586 pwmnd 18951 cutlt 27967 clsk1indlem3 44061 isotone1 44066 isgrtri 47915 | 
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