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Theorem pwuni 4189
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 3835 . . 3  |-  ( x  e.  A  ->  x  C_ 
U. A )
2 vex 2740 . . . 4  |-  x  e. 
_V
32elpw 3580 . . 3  |-  ( x  e.  ~P U. A  <->  x 
C_  U. A )
41, 3sylibr 134 . 2  |-  ( x  e.  A  ->  x  e.  ~P U. A )
54ssriv 3159 1  |-  A  C_  ~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 2148    C_ wss 3129   ~Pcpw 3574   U.cuni 3807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576  df-uni 3808
This theorem is referenced by:  uniexb  4469  2pwuninelg  6277  istopon  13144  eltg3i  13189  mopnfss  13580
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