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Theorem unipwr 29019
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4417. The proof of this theorem was automatically generated from unipwrVD 29018 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 2961 . . . 4  |-  x  e. 
_V
21snid 3843 . . 3  |-  x  e. 
{ x }
3 snelpwi 4412 . . 3  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 elunii 4022 . . 3  |-  ( ( x  e.  { x }  /\  { x }  e.  ~P A )  ->  x  e.  U. ~P A
)
52, 3, 4sylancr 646 . 2  |-  ( x  e.  A  ->  x  e.  U. ~P A )
65ssriv 3354 1  |-  A  C_  U. ~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1726    C_ wss 3322   ~Pcpw 3801   {csn 3816   U.cuni 4017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-pr 3823  df-uni 4018
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