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Theorem pwuninel2 6308
Description: The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
pwuninel2  |-  ( U. A  e.  V  ->  -. 
~P U. A  e.  A
)

Proof of Theorem pwuninel2
StepHypRef Expression
1 pwnss 4177 . 2  |-  ( U. A  e.  V  ->  -. 
~P U. A  C_  U. A
)
2 elssuni 3852 . 2  |-  ( ~P
U. A  e.  A  ->  ~P U. A  C_  U. A )
31, 2nsyl 629 1  |-  ( U. A  e.  V  ->  -. 
~P U. A  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 2160    C_ wss 3144   ~Pcpw 3590   U.cuni 3824
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-nel 2456  df-rab 2477  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-uni 3825
This theorem is referenced by:  pnfnre  8030
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