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Theorem unipw 4053
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.)
Assertion
Ref Expression
unipw 𝒫 𝐴 = 𝐴

Proof of Theorem unipw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3662 . . . 4 (𝑥 𝒫 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴))
2 elelpwi 3445 . . . . 5 ((𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
32exlimiv 1535 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
41, 3sylbi 120 . . 3 (𝑥 𝒫 𝐴𝑥𝐴)
5 vex 2623 . . . . 5 𝑥 ∈ V
65snid 3479 . . . 4 𝑥 ∈ {𝑥}
7 snelpwi 4048 . . . 4 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
8 elunii 3664 . . . 4 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
96, 7, 8sylancr 406 . . 3 (𝑥𝐴𝑥 𝒫 𝐴)
104, 9impbii 125 . 2 (𝑥 𝒫 𝐴𝑥𝐴)
1110eqriv 2086 1 𝒫 𝐴 = 𝐴
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1290  wex 1427  wcel 1439  𝒫 cpw 3433  {csn 3450   cuni 3659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-uni 3660
This theorem is referenced by:  pwtr  4055  pwexb  4309  univ  4311  unixpss  4564  eltg4i  11809  distop  11839  distopon  11841  distps  11845  ntrss2  11875  isopn3  11879  discld  11890
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