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Theorem 2pwuninelg 6183
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 4472 . 2  |-  -.  ( A  e.  ~P ~P U. A  /\  ~P ~P U. A  e.  A )
2 pwuni 4119 . . . 4  |-  A  C_  ~P U. A
3 elpwg 3518 . . . 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 653 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 1480    C_ wss 3071   ~Pcpw 3510   U.cuni 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4455
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3740
This theorem is referenced by:  mnfnre  7827
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