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Theorem unipw 4235
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 3827 . . . 4  |-  ( x  e.  U. ~P A  <->  E. y ( x  e.  y  /\  y  e. 
~P A ) )
2 elelpwi 3602 . . . . 5  |-  ( ( x  e.  y  /\  y  e.  ~P A
)  ->  x  e.  A )
32exlimiv 1609 . . . 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 2755 . . . . 5  |-  x  e. 
_V
65snid 3638 . . . 4  |-  x  e. 
{ x }
7 snelpwi 4230 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
8 elunii 3829 . . . 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 2186 1  |-  U. ~P A  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364   E.wex 1503    e. wcel 2160   ~Pcpw 3590   {csn 3607   U.cuni 3824
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-uni 3825
This theorem is referenced by:  pwtr  4237  pwexb  4492  univ  4494  unixpss  4757  eltg4i  14015  distop  14045  distopon  14047  distps  14051  ntrss2  14081  isopn3  14085  discld  14096  txdis  14237
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