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Theorem pwuni 4626
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 4224 . . 3  |-  ( x  e.  A  ->  x  C_ 
U. A )
2 selpw 3970 . . 3  |-  ( x  e.  ~P U. A  <->  x 
C_  U. A )
31, 2sylibr 212 . 2  |-  ( x  e.  A  ->  x  e.  ~P U. A )
43ssriv 3463 1  |-  A  C_  ~P U. A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758    C_ wss 3431   ~Pcpw 3963   U.cuni 4194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-v 3074  df-in 3438  df-ss 3445  df-pw 3965  df-uni 4195
This theorem is referenced by:  uniexb  6491  fipwuni  7782  uniwf  8132  rankuni  8176  rankc2  8184  rankxplim  8192  fin23lem17  8613  axcclem  8732  grurn  9074  istopon  18657  eltg3i  18693  cmpfi  19138  hmphdis  19496  ptcmpfi  19513  fbssfi  19537  mopnfss  20145  shsspwh  24796  hasheuni  26674  issgon  26706  sigaclci  26715  sigagenval  26723  dmsigagen  26727  imambfm  26816
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