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Theorem 2pwuninelg 5839
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  V  ~P ~P U.

Proof of Theorem 2pwuninelg
StepHypRef Expression
1 en2lp 4232 . 2  ~P ~P U.  ~P ~P U.
2 pwuni 3934 . . . 4  C_  ~P U.
3 elpwg 3359 . . . 4  V  ~P ~P U.  C_  ~P U.
42, 3mpbiri 157 . . 3  V  ~P ~P U.
5 ax-ia3 101 . . 3  ~P ~P U.  ~P ~P U.  ~P ~P U.  ~P ~P U.
64, 5syl 14 . 2  V  ~P ~P U.  ~P ~P U.  ~P ~P U.
71, 6mtoi 589 1  V  ~P ~P U.
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wcel 1390    C_ wss 2911   ~Pcpw 3351   U.cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by:  mnfnre  6865
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