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Theorem 2pwuninelg 5932
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 4305 . 2  |-  -.  ( A  e.  ~P ~P U. A  /\  ~P ~P U. A  e.  A )
2 pwuni 3971 . . . 4  |-  A  C_  ~P U. A
3 elpwg 3398 . . . 4  |-  ( A  e.  V  ->  ( A  e.  ~P ~P U. A  <->  A  C_  ~P U. A ) )
42, 3mpbiri 166 . . 3  |-  ( A  e.  V  ->  A  e.  ~P ~P U. A
)
5 ax-ia3 106 . . 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 623 1  |-  ( A  e.  V  ->  -.  ~P ~P U. A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    e. wcel 1434    C_ wss 2974   ~Pcpw 3390   U.cuni 3609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610
This theorem is referenced by:  mnfnre  7223
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