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Theorem unipw 4687
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 4238 . . . 4  |-  ( x  e.  U. ~P A  <->  E. y ( x  e.  y  /\  y  e. 
~P A ) )
2 elelpwi 4010 . . . . 5  |-  ( ( x  e.  y  /\  y  e.  ~P A
)  ->  x  e.  A )
32exlimiv 1727 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e. 
~P A )  ->  x  e.  A )
41, 3sylbi 195 . . 3  |-  ( x  e.  U. ~P A  ->  x  e.  A )
5 ssnid 4045 . . . 4  |-  x  e. 
{ x }
6 snelpwi 4682 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
7 elunii 4240 . . . 4  |-  ( ( x  e.  { x }  /\  { x }  e.  ~P A )  ->  x  e.  U. ~P A
)
85, 6, 7sylancr 661 . . 3  |-  ( x  e.  A  ->  x  e.  U. ~P A )
94, 8impbii 188 . 2  |-  ( x  e.  U. ~P A  <->  x  e.  A )
109eqriv 2450 1  |-  U. ~P A  =  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   ~Pcpw 3999   {csn 4016   U.cuni 4235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001  df-sn 4017  df-pr 4019  df-uni 4236
This theorem is referenced by:  univ  4688  pwtr  4690  unixpss  5106  pwexb  6584  unifpw  7815  fiuni  7880  ween  8407  fin23lem41  8723  mremre  15093  submre  15094  isacs1i  15146  eltg4i  19628  distop  19664  distopon  19665  distps  19683  ntrss2  19725  isopn3  19734  discld  19757  mretopd  19760  dishaus  20050  discmp  20065  dissnlocfin  20196  locfindis  20197  txdis  20299  xkopt  20322  xkofvcn  20351  hmphdis  20463  ustbas2  20894  vitali  22188  shsupcl  26454  shsupunss  26462  iundifdifd  27639  iundifdif  27640  dispcmp  28097  mbfmcnt  28476  omssubadd  28508  carsgval  28511  carsggect  28526  coinflipprob  28682  coinflipuniv  28684  fnemeet2  30425  ismrcd1  30870  hbt  31320  mapdunirnN  37774  pwelg  38158
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