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Theorem 2pwuninelg 6307
Description: The power set of 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 Jim Kingdon, 14-Jan-2020.)
Assertion
Ref Expression
2pwuninelg  |-  ( A  e.  V  ->  -.  ~P ~P U. A  e.  A )

Proof of Theorem 2pwuninelg
StepHypRef Expression
1 en2lp 4571 . 2  |-  -.  ( A  e.  ~P ~P U. A  /\  ~P ~P U. A  e.  A )
2 pwuni 4210 . . . 4  |-  A  C_  ~P U. A
3 elpwg 3598 . . . 4  |-  ( A  e.  V  ->  ( A  e.  ~P ~P U. A  <->  A  C_  ~P U. A ) )
42, 3mpbiri 168 . . 3  |-  ( A  e.  V  ->  A  e.  ~P ~P U. A
)
5 ax-ia3 108 . . 3  |-  ( A  e.  ~P ~P U. A  ->  ( ~P ~P U. A  e.  A  -> 
( A  e.  ~P ~P U. A  /\  ~P ~P U. A  e.  A
) ) )
64, 5syl 14 . 2  |-  ( A  e.  V  ->  ( ~P ~P U. A  e.  A  ->  ( A  e.  ~P ~P U. A  /\  ~P ~P U. A  e.  A ) ) )
71, 6mtoi 665 1  |-  ( A  e.  V  ->  -.  ~P ~P U. A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    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-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825
This theorem is referenced by:  mnfnre  8029
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