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Theorem pwuni 4100
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
StepHypRef Expression
1 elssuni 3753 . . 3  |-  ( x  e.  A  ->  x  C_ 
U. A )
2 vex 2730 . . . 4  |-  x  e. 
_V
32elpw 3536 . . 3  |-  ( x  e.  ~P U. A  <->  x 
C_  U. A )
41, 3sylibr 205 . 2  |-  ( x  e.  A  ->  x  e.  ~P U. A )
54ssriv 3105 1  |-  A  C_  ~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 1621    C_ wss 3078   ~Pcpw 3530   U.cuni 3727
This theorem is referenced by:  uniexb  4454  fipwuni  7063  uniwf  7375  rankuni  7419  rankc2  7427  rankxplim  7433  fin23lem17  7848  axcclem  7967  grurn  8303  istopon  16495  eltg3i  16531  cmpfi  16967  hmphdis  17319  ptcmpfi  17336  fbssfi  17364  mopnfss  17821  shsspwh  21655  unfinsef  24234
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-in 3085  df-ss 3089  df-pw 3532  df-uni 3728
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