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Theorem unipwr 28361
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4327. The proof of this theorem was automatically generated from unipwrVD 28360 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
unipwr  |-  A  C_  U. ~P A

Proof of Theorem unipwr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2876 . . . 4  |-  x  e. 
_V
21snid 3756 . . 3  |-  x  e. 
{ x }
3 snelpwi 4322 . . 3  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 elunii 3934 . . 3  |-  ( ( x  e.  { x }  /\  { x }  e.  ~P A )  ->  x  e.  U. ~P A
)
52, 3, 4sylancr 644 . 2  |-  ( x  e.  A  ->  x  e.  U. ~P A )
65ssriv 3270 1  |-  A  C_  U. ~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1715    C_ wss 3238   ~Pcpw 3714   {csn 3729   U.cuni 3929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-pw 3716  df-sn 3735  df-pr 3736  df-uni 3930
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