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Theorem unipw 4214
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 3810 . . . 4  |-  ( x  e.  U. ~P A  <->  E. y ( x  e.  y  /\  y  e. 
~P A ) )
2 elelpwi 3586 . . . . 5  |-  ( ( x  e.  y  /\  y  e.  ~P A
)  ->  x  e.  A )
32exlimiv 1598 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e. 
~P A )  ->  x  e.  A )
41, 3sylbi 121 . . 3  |-  ( x  e.  U. ~P A  ->  x  e.  A )
5 vex 2740 . . . . 5  |-  x  e. 
_V
65snid 3622 . . . 4  |-  x  e. 
{ x }
7 snelpwi 4209 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
8 elunii 3812 . . . 4  |-  ( ( x  e.  { x }  /\  { x }  e.  ~P A )  ->  x  e.  U. ~P A
)
96, 7, 8sylancr 414 . . 3  |-  ( x  e.  A  ->  x  e.  U. ~P A )
104, 9impbii 126 . 2  |-  ( x  e.  U. ~P A  <->  x  e.  A )
1110eqriv 2174 1  |-  U. ~P A  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353   E.wex 1492    e. wcel 2148   ~Pcpw 3574   {csn 3591   U.cuni 3807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-uni 3808
This theorem is referenced by:  pwtr  4216  pwexb  4471  univ  4473  unixpss  4736  eltg4i  13222  distop  13252  distopon  13254  distps  13258  ntrss2  13288  isopn3  13292  discld  13303  txdis  13444
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