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Theorem unipw 4107
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 3707 . . . 4 (𝑥 𝒫 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴))
2 elelpwi 3490 . . . . 5 ((𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
32exlimiv 1560 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
41, 3sylbi 120 . . 3 (𝑥 𝒫 𝐴𝑥𝐴)
5 vex 2661 . . . . 5 𝑥 ∈ V
65snid 3524 . . . 4 𝑥 ∈ {𝑥}
7 snelpwi 4102 . . . 4 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
8 elunii 3709 . . . 4 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
96, 7, 8sylancr 408 . . 3 (𝑥𝐴𝑥 𝒫 𝐴)
104, 9impbii 125 . 2 (𝑥 𝒫 𝐴𝑥𝐴)
1110eqriv 2112 1 𝒫 𝐴 = 𝐴
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wex 1451  wcel 1463  𝒫 cpw 3478  {csn 3495   cuni 3704
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-uni 3705
This theorem is referenced by:  pwtr  4109  pwexb  4363  univ  4365  unixpss  4620  eltg4i  12119  distop  12149  distopon  12151  distps  12155  ntrss2  12185  isopn3  12189  discld  12200  txdis  12341
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