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Theorem unipw 4149
 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

Proof of Theorem unipw
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3748 . . . 4
2 elelpwi 3528 . . . . 5
32exlimiv 1578 . . . 4
41, 3sylbi 120 . . 3
5 vex 2693 . . . . 5
65snid 3564 . . . 4
7 snelpwi 4144 . . . 4
8 elunii 3750 . . . 4
96, 7, 8sylancr 411 . . 3
104, 9impbii 125 . 2
1110eqriv 2137 1
 Colors of variables: wff set class Syntax hints:   wa 103   wceq 1332  wex 1469   wcel 1481  cpw 3516  csn 3533  cuni 3745 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 4055  ax-pow 4107 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 2692  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-uni 3746 This theorem is referenced by:  pwtr  4151  pwexb  4405  univ  4407  unixpss  4663  eltg4i  12297  distop  12327  distopon  12329  distps  12333  ntrss2  12363  isopn3  12367  discld  12378  txdis  12519
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