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Theorem pwuni 4336
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 3985 . . 3  |-  ( x  e.  A  ->  x  C_ 
U. A )
2 vex 2902 . . . 4  |-  x  e. 
_V
32elpw 3748 . . 3  |-  ( x  e.  ~P U. A  <->  x 
C_  U. A )
41, 3sylibr 204 . 2  |-  ( x  e.  A  ->  x  e.  ~P U. A )
54ssriv 3295 1  |-  A  C_  ~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 1717    C_ wss 3263   ~Pcpw 3742   U.cuni 3957
This theorem is referenced by:  uniexb  4692  fipwuni  7366  uniwf  7678  rankuni  7722  rankc2  7730  rankxplim  7736  fin23lem17  8151  axcclem  8270  grurn  8609  istopon  16913  eltg3i  16949  cmpfi  17393  hmphdis  17749  ptcmpfi  17766  fbssfi  17790  mopnfss  18363  shsspwh  22596  hasheuni  24271  issgon  24302  sigaclci  24311  sigagenval  24319  dmsigagen  24323  imambfm  24406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-in 3270  df-ss 3277  df-pw 3744  df-uni 3958
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