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Theorem unipwr 28609
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4224. The proof of this theorem was automatically generated from unipwrVD 28608 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 2791 . . . 4  |-  x  e. 
_V
21snid 3667 . . 3  |-  x  e. 
{ x }
3 snelpwi 4220 . . 3  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 elunii 3832 . . 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 3184 1  |-  A  C_  U. ~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828
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