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Theorem unipw 3953
 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 3583 . . . 4
2 elelpwi 3370 . . . . 5
32exlimiv 1489 . . . 4
41, 3sylbi 114 . . 3
5 vex 2560 . . . . 5
65snid 3402 . . . 4
7 snelpwi 3948 . . . 4
8 elunii 3585 . . . 4
96, 7, 8sylancr 393 . . 3
104, 9impbii 117 . 2
1110eqriv 2037 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1243  wex 1381   wcel 1393  cpw 3359  csn 3375  cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-uni 3581 This theorem is referenced by:  pwtr  3955  pwexb  4206  univ  4207  unixpss  4451
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