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Theorem pwuni 4678
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 4275 . . 3  |-  ( x  e.  A  ->  x  C_ 
U. A )
2 selpw 4017 . . 3  |-  ( x  e.  ~P U. A  <->  x 
C_  U. A )
31, 2sylibr 212 . 2  |-  ( x  e.  A  ->  x  e.  ~P U. A )
43ssriv 3508 1  |-  A  C_  ~P U. A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767    C_ wss 3476   ~Pcpw 4010   U.cuni 4245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-ss 3490  df-pw 4012  df-uni 4246
This theorem is referenced by:  uniexb  6588  fipwuni  7882  uniwf  8233  rankuni  8277  rankc2  8285  rankxplim  8293  fin23lem17  8714  axcclem  8833  grurn  9175  istopon  19190  eltg3i  19226  cmpfi  19671  hmphdis  20029  ptcmpfi  20046  fbssfi  20070  mopnfss  20678  shsspwh  25837  circtopn  27635  hasheuni  27728  issgon  27760  sigaclci  27769  sigagenval  27777  dmsigagen  27781  imambfm  27870
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