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Theorem unipwr 28663
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4382. The proof of this theorem was automatically generated from unipwrVD 28662 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 2927 . . . 4  |-  x  e. 
_V
21snid 3809 . . 3  |-  x  e. 
{ x }
3 snelpwi 4377 . . 3  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 elunii 3988 . . 3  |-  ( ( x  e.  { x }  /\  { x }  e.  ~P A )  ->  x  e.  U. ~P A
)
52, 3, 4sylancr 645 . 2  |-  ( x  e.  A  ->  x  e.  U. ~P A )
65ssriv 3320 1  |-  A  C_  U. ~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1721    C_ wss 3288   ~Pcpw 3767   {csn 3782   U.cuni 3983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-pw 3769  df-sn 3788  df-pr 3789  df-uni 3984
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