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Theorem unipw 3916
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

Proof of Theorem unipw
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3546 . . . 4  U. ~P 
~P
2 elelpwi 3334 . . . . 5  ~P
32exlimiv 1462 . . . 4  ~P
41, 3sylbi 114 . . 3  U. ~P
5 vex 2529 . . . . 5 
_V
65snid 3366 . . . 4 
{ }
7 snelpwi 3911 . . . 4  { }  ~P
8 elunii 3548 . . . 4  { }  { }  ~P  U. ~P
96, 7, 8sylancr 393 . . 3  U. ~P
104, 9impbii 117 . 2  U. ~P
1110eqriv 2010 1  U. ~P
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1223  wex 1354   wcel 1366   ~Pcpw 3323   {csn 3339   U.cuni 3543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-uni 3544
This theorem is referenced by:  pwtr  3918  pwexb  4144  univ  4145  unixpss  4366
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