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Theorem 2pwuninel 7665
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 NM, 27-Jun-2008.)
Assertion
Ref Expression
2pwuninel  |-  -.  ~P ~P U. A  e.  A

Proof of Theorem 2pwuninel
StepHypRef Expression
1 sdomirr 7647 . . 3  |-  -.  ~P ~P U. A  ~<  ~P ~P U. A
2 elssuni 4264 . . . 4  |-  ( ~P ~P U. A  e.  A  ->  ~P ~P U. A  C_  U. A )
3 ssdomg 7554 . . . . 5  |-  ( U. A  e.  _V  ->  ( ~P ~P U. A  C_ 
U. A  ->  ~P ~P U. A  ~<_  U. A
) )
4 canth2g 7664 . . . . . 6  |-  ( U. A  e.  _V  ->  U. A  ~<  ~P U. A
)
5 pwexb 6584 . . . . . . 7  |-  ( U. A  e.  _V  <->  ~P U. A  e.  _V )
6 canth2g 7664 . . . . . . 7  |-  ( ~P
U. A  e.  _V  ->  ~P U. A  ~<  ~P ~P U. A )
75, 6sylbi 195 . . . . . 6  |-  ( U. A  e.  _V  ->  ~P
U. A  ~<  ~P ~P U. A )
8 sdomtr 7648 . . . . . 6  |-  ( ( U. A  ~<  ~P U. A  /\  ~P U. A  ~<  ~P ~P U. A
)  ->  U. A  ~<  ~P ~P U. A )
94, 7, 8syl2anc 659 . . . . 5  |-  ( U. A  e.  _V  ->  U. A  ~<  ~P ~P U. A )
10 domsdomtr 7645 . . . . . 6  |-  ( ( ~P ~P U. A  ~<_  U. A  /\  U. A  ~<  ~P ~P U. A
)  ->  ~P ~P U. A  ~<  ~P ~P U. A )
1110ex 432 . . . . 5  |-  ( ~P ~P U. A  ~<_  U. A  ->  ( U. A  ~<  ~P ~P U. A  ->  ~P ~P U. A  ~<  ~P ~P U. A ) )
123, 9, 11syl6ci 65 . . . 4  |-  ( U. A  e.  _V  ->  ( ~P ~P U. A  C_ 
U. A  ->  ~P ~P U. A  ~<  ~P ~P U. A ) )
132, 12syl5 32 . . 3  |-  ( U. A  e.  _V  ->  ( ~P ~P U. A  e.  A  ->  ~P ~P U. A  ~<  ~P ~P U. A ) )
141, 13mtoi 178 . 2  |-  ( U. A  e.  _V  ->  -. 
~P ~P U. A  e.  A )
15 elex 3115 . . . 4  |-  ( ~P ~P U. A  e.  A  ->  ~P ~P U. A  e.  _V )
16 pwexb 6584 . . . . 5  |-  ( ~P
U. A  e.  _V  <->  ~P ~P U. A  e. 
_V )
175, 16bitri 249 . . . 4  |-  ( U. A  e.  _V  <->  ~P ~P U. A  e.  _V )
1815, 17sylibr 212 . . 3  |-  ( ~P ~P U. A  e.  A  ->  U. A  e. 
_V )
1918con3i 135 . 2  |-  ( -. 
U. A  e.  _V  ->  -.  ~P ~P U. A  e.  A )
2014, 19pm2.61i 164 1  |-  -.  ~P ~P U. A  e.  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1823   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   class class class wbr 4439    ~<_ cdom 7507    ~< csdm 7508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512
This theorem is referenced by:  mnfnre  9625
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