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Theorem pwuni 4355
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 4003 . . 3  |-  ( x  e.  A  ->  x  C_ 
U. A )
2 vex 2919 . . . 4  |-  x  e. 
_V
32elpw 3765 . . 3  |-  ( x  e.  ~P U. A  <->  x 
C_  U. A )
41, 3sylibr 204 . 2  |-  ( x  e.  A  ->  x  e.  ~P U. A )
54ssriv 3312 1  |-  A  C_  ~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 1721    C_ wss 3280   ~Pcpw 3759   U.cuni 3975
This theorem is referenced by:  uniexb  4711  fipwuni  7389  uniwf  7701  rankuni  7745  rankc2  7753  rankxplim  7759  fin23lem17  8174  axcclem  8293  grurn  8632  istopon  16945  eltg3i  16981  cmpfi  17425  hmphdis  17781  ptcmpfi  17798  fbssfi  17822  mopnfss  18426  shsspwh  22701  hasheuni  24428  issgon  24459  sigaclci  24468  sigagenval  24476  dmsigagen  24480  imambfm  24565
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287  df-ss 3294  df-pw 3761  df-uni 3976
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