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Theorem pwuni 4206
Description: A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
pwuni  |-  A  C_  ~P U. A

Proof of Theorem pwuni
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elssuni 3855 . . 3  |-  ( x  e.  A  ->  x  C_ 
U. A )
2 vex 2791 . . . 4  |-  x  e. 
_V
32elpw 3631 . . 3  |-  ( x  e.  ~P U. A  <->  x 
C_  U. A )
41, 3sylibr 203 . 2  |-  ( x  e.  A  ->  x  e.  ~P U. A )
54ssriv 3184 1  |-  A  C_  ~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827
This theorem is referenced by:  uniexb  4563  fipwuni  7179  uniwf  7491  rankuni  7535  rankc2  7543  rankxplim  7549  fin23lem17  7964  axcclem  8083  grurn  8423  istopon  16663  eltg3i  16699  cmpfi  17135  hmphdis  17487  ptcmpfi  17504  fbssfi  17532  mopnfss  17989  shsspwh  21825  hasheuni  23453  issgon  23484  sigaclci  23493  sigagenval  23501  dmsigagen  23505  imambfm  23567  unfinsef  25069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828
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