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Mirrors > Home > ILE Home > Th. List > pwssunim | GIF version |
Description: The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.) |
Ref | Expression |
---|---|
pwssunim | ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn2 3254 | . . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴) | |
2 | pweq 3518 | . . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → 𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐴) | |
3 | eqimss 3156 | . . . . . 6 ⊢ (𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐴 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴) |
5 | 1, 4 | sylbi 120 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴) |
6 | ssequn1 3251 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) | |
7 | pweq 3518 | . . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → 𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐵) | |
8 | eqimss 3156 | . . . . . 6 ⊢ (𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐵 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵) | |
9 | 7, 8 | syl 14 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵) |
10 | 6, 9 | sylbi 120 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵) |
11 | 5, 10 | orim12i 749 | . . 3 ⊢ ((𝐵 ⊆ 𝐴 ∨ 𝐴 ⊆ 𝐵) → (𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵)) |
12 | 11 | orcoms 720 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → (𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵)) |
13 | ssun 3260 | . 2 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵)) | |
14 | 12, 13 | syl 14 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 = wceq 1332 ∪ cun 3074 ⊆ wss 3076 𝒫 cpw 3515 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 |
This theorem is referenced by: pwunim 4216 |
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