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Theorem unipw 3944
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  U. ~P

Proof of Theorem unipw
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3574 . . . 4  U. ~P 
~P
2 elelpwi 3362 . . . . 5  ~P
32exlimiv 1486 . . . 4  ~P
41, 3sylbi 114 . . 3  U. ~P
5 vex 2554 . . . . 5 
_V
65snid 3394 . . . 4 
{ }
7 snelpwi 3939 . . . 4  { }  ~P
8 elunii 3576 . . . 4  { }  { }  ~P  U. ~P
96, 7, 8sylancr 393 . . 3  U. ~P
104, 9impbii 117 . 2  U. ~P
1110eqriv 2034 1  U. ~P
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1242  wex 1378   wcel 1390   ~Pcpw 3351   {csn 3367   U.cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-uni 3572
This theorem is referenced by:  pwtr  3946  pwexb  4172  univ  4173  unixpss  4394
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