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Mirrors > Home > ILE Home > Th. List > unipw | GIF version |
Description: A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.) |
Ref | Expression |
---|---|
unipw | ⊢ ∪ 𝒫 𝐴 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 3583 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)) | |
2 | elelpwi 3370 | . . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴) | |
3 | 2 | exlimiv 1489 | . . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴) |
4 | 1, 3 | sylbi 114 | . . 3 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 → 𝑥 ∈ 𝐴) |
5 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | snid 3402 | . . . 4 ⊢ 𝑥 ∈ {𝑥} |
7 | snelpwi 3948 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
8 | elunii 3585 | . . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴) | |
9 | 6, 7, 8 | sylancr 393 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴) |
10 | 4, 9 | impbii 117 | . 2 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴) |
11 | 10 | eqriv 2037 | 1 ⊢ ∪ 𝒫 𝐴 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 𝒫 cpw 3359 {csn 3375 ∪ cuni 3580 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-uni 3581 |
This theorem is referenced by: pwtr 3955 pwexb 4206 univ 4207 unixpss 4451 |
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