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Theorem unipw 4008
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 A  =  A

Proof of Theorem unipw
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3630 . . . 4  |-  ( x  e.  U. ~P A  <->  E. y ( x  e.  y  /\  y  e. 
~P A ) )
2 elelpwi 3417 . . . . 5  |-  ( ( x  e.  y  /\  y  e.  ~P A
)  ->  x  e.  A )
32exlimiv 1530 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e. 
~P A )  ->  x  e.  A )
41, 3sylbi 119 . . 3  |-  ( x  e.  U. ~P A  ->  x  e.  A )
5 vex 2615 . . . . 5  |-  x  e. 
_V
65snid 3449 . . . 4  |-  x  e. 
{ x }
7 snelpwi 4003 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
8 elunii 3632 . . . 4  |-  ( ( x  e.  { x }  /\  { x }  e.  ~P A )  ->  x  e.  U. ~P A
)
96, 7, 8sylancr 405 . . 3  |-  ( x  e.  A  ->  x  e.  U. ~P A )
104, 9impbii 124 . 2  |-  ( x  e.  U. ~P A  <->  x  e.  A )
1110eqriv 2080 1  |-  U. ~P A  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   ~Pcpw 3406   {csn 3422   U.cuni 3627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-uni 3628
This theorem is referenced by:  pwtr  4010  pwexb  4260  univ  4261  unixpss  4509
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