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Theorem 2pwuninelg 6110
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 4407 . 2  |-  -.  ( A  e.  ~P ~P U. A  /\  ~P ~P U. A  e.  A )
2 pwuni 4056 . . . 4  |-  A  C_  ~P U. A
3 elpwg 3465 . . . 4  |-  ( A  e.  V  ->  ( A  e.  ~P ~P U. A  <->  A  C_  ~P U. A ) )
42, 3mpbiri 167 . . 3  |-  ( A  e.  V  ->  A  e.  ~P ~P U. A
)
5 ax-ia3 107 . . 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 631 1  |-  ( A  e.  V  ->  -.  ~P ~P U. A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    e. wcel 1448    C_ wss 3021   ~Pcpw 3457   U.cuni 3683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684
This theorem is referenced by:  mnfnre  7680
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