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Theorem unipw 4109
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 3709 . . . 4 (𝑥 𝒫 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴))
2 elelpwi 3492 . . . . 5 ((𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
32exlimiv 1562 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
41, 3sylbi 120 . . 3 (𝑥 𝒫 𝐴𝑥𝐴)
5 vex 2663 . . . . 5 𝑥 ∈ V
65snid 3526 . . . 4 𝑥 ∈ {𝑥}
7 snelpwi 4104 . . . 4 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
8 elunii 3711 . . . 4 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
96, 7, 8sylancr 410 . . 3 (𝑥𝐴𝑥 𝒫 𝐴)
104, 9impbii 125 . 2 (𝑥 𝒫 𝐴𝑥𝐴)
1110eqriv 2114 1 𝒫 𝐴 = 𝐴
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wex 1453  wcel 1465  𝒫 cpw 3480  {csn 3497   cuni 3706
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-uni 3707
This theorem is referenced by:  pwtr  4111  pwexb  4365  univ  4367  unixpss  4622  eltg4i  12151  distop  12181  distopon  12183  distps  12187  ntrss2  12217  isopn3  12221  discld  12232  txdis  12373
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