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Theorem unipw 4147
 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 3747 . . . 4 (𝑥 𝒫 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴))
2 elelpwi 3527 . . . . 5 ((𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
32exlimiv 1578 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
41, 3sylbi 120 . . 3 (𝑥 𝒫 𝐴𝑥𝐴)
5 vex 2692 . . . . 5 𝑥 ∈ V
65snid 3563 . . . 4 𝑥 ∈ {𝑥}
7 snelpwi 4142 . . . 4 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
8 elunii 3749 . . . 4 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
96, 7, 8sylancr 411 . . 3 (𝑥𝐴𝑥 𝒫 𝐴)
104, 9impbii 125 . 2 (𝑥 𝒫 𝐴𝑥𝐴)
1110eqriv 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  12264  distop  12294  distopon  12296  distps  12300  ntrss2  12330  isopn3  12334  discld  12345  txdis  12486
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