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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 3747 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)) | |
2 | elelpwi 3527 | . . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴) | |
3 | 2 | exlimiv 1578 | . . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴) |
4 | 1, 3 | sylbi 120 | . . 3 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 → 𝑥 ∈ 𝐴) |
5 | vex 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | snid 3563 | . . . 4 ⊢ 𝑥 ∈ {𝑥} |
7 | snelpwi 4142 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
8 | elunii 3749 | . . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴) | |
9 | 6, 7, 8 | sylancr 411 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴) |
10 | 4, 9 | impbii 125 | . 2 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴) |
11 | 10 | eqriv 2137 | 1 ⊢ ∪ 𝒫 𝐴 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 𝒫 cpw 3515 {csn 3532 ∪ cuni 3744 |
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-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 |
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-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-uni 3745 |
This theorem is referenced by: pwtr 4149 pwexb 4403 univ 4405 unixpss 4660 eltg4i 12263 distop 12293 distopon 12295 distps 12299 ntrss2 12329 isopn3 12333 discld 12344 txdis 12485 |
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