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Mirrors > Home > ILE Home > Th. List > pwunss | 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.) |
Ref | Expression |
---|---|
pwunss | ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun 3286 | . . 3 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐴 ∪ 𝐵)) | |
2 | elun 3248 | . . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵)) | |
3 | vex 2715 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | 3 | elpw 3549 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
5 | 3 | elpw 3549 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) |
6 | 4, 5 | orbi12i 754 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)) |
7 | 2, 6 | bitri 183 | . . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)) |
8 | 3 | elpw 3549 | . . 3 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∪ 𝐵)) |
9 | 1, 7, 8 | 3imtr4i 200 | . 2 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) → 𝑥 ∈ 𝒫 (𝐴 ∪ 𝐵)) |
10 | 9 | ssriv 3132 | 1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 698 ∈ wcel 2128 ∪ cun 3100 ⊆ wss 3102 𝒫 cpw 3543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 |
This theorem is referenced by: pwundifss 4245 pwunim 4246 |
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