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Theorem pwuninel2 5897
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 3912 . 2  |-  ( U. A  e.  V  ->  -. 
~P U. A  C_  U. A
)
2 elssuni 3608 . 2  |-  ( ~P
U. A  e.  A  ->  ~P U. A  C_  U. A )
31, 2nsyl 558 1  |-  ( U. A  e.  V  ->  -. 
~P U. A  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1393    C_ wss 2917   ~Pcpw 3359   U.cuni 3580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581
This theorem is referenced by:  pnfnre  7065
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