Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3786 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)) | |
2 | elelpwi 3565 | . . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴) | |
3 | 2 | exlimiv 1585 | . . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴) |
4 | 1, 3 | sylbi 120 | . . 3 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 → 𝑥 ∈ 𝐴) |
5 | vex 2724 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | snid 3601 | . . . 4 ⊢ 𝑥 ∈ {𝑥} |
7 | snelpwi 4184 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
8 | elunii 3788 | . . . 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 2161 | 1 ⊢ ∪ 𝒫 𝐴 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∃wex 1479 ∈ wcel 2135 𝒫 cpw 3553 {csn 3570 ∪ cuni 3783 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-uni 3784 |
This theorem is referenced by: pwtr 4191 pwexb 4446 univ 4448 unixpss 4711 eltg4i 12602 distop 12632 distopon 12634 distps 12638 ntrss2 12668 isopn3 12672 discld 12683 txdis 12824 |
Copyright terms: Public domain | W3C validator |