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Theorem unipw 4097
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 3703 . . . 4  |-  ( x  e.  U. ~P A  <->  E. y ( x  e.  y  /\  y  e. 
~P A ) )
2 elelpwi 3486 . . . . 5  |-  ( ( x  e.  y  /\  y  e.  ~P A
)  ->  x  e.  A )
32exlimiv 1558 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e. 
~P A )  ->  x  e.  A )
41, 3sylbi 120 . . 3  |-  ( x  e.  U. ~P A  ->  x  e.  A )
5 vex 2658 . . . . 5  |-  x  e. 
_V
65snid 3520 . . . 4  |-  x  e. 
{ x }
7 snelpwi 4092 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
8 elunii 3705 . . . 4  |-  ( ( x  e.  { x }  /\  { x }  e.  ~P A )  ->  x  e.  U. ~P A
)
96, 7, 8sylancr 408 . . 3  |-  ( x  e.  A  ->  x  e.  U. ~P A )
104, 9impbii 125 . 2  |-  ( x  e.  U. ~P A  <->  x  e.  A )
1110eqriv 2110 1  |-  U. ~P A  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1312   E.wex 1449    e. wcel 1461   ~Pcpw 3474   {csn 3491   U.cuni 3700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-uni 3701
This theorem is referenced by:  pwtr  4099  pwexb  4353  univ  4355  unixpss  4610  eltg4i  12061  distop  12091  distopon  12093  distps  12097  ntrss2  12127  isopn3  12131  discld  12142  txdis  12282
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