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Theorem 2pwne 7675
Description: No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
Assertion
Ref Expression
2pwne  |-  ( A  e.  V  ->  ~P ~P A  =/=  A
)

Proof of Theorem 2pwne
StepHypRef Expression
1 sdomirr 7656 . . 3  |-  -.  ~P ~P A  ~<  ~P ~P A
2 canth2g 7673 . . . . 5  |-  ( A  e.  V  ->  A  ~<  ~P A )
3 pwexg 4621 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
4 canth2g 7673 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  ~<  ~P ~P A )
53, 4syl 16 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~<  ~P ~P A
)
6 sdomtr 7657 . . . . 5  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<  ~P ~P A )  ->  A  ~<  ~P ~P A )
72, 5, 6syl2anc 661 . . . 4  |-  ( A  e.  V  ->  A  ~<  ~P ~P A )
8 breq1 4440 . . . 4  |-  ( ~P ~P A  =  A  ->  ( ~P ~P A  ~<  ~P ~P A  <->  A 
~<  ~P ~P A ) )
97, 8syl5ibrcom 222 . . 3  |-  ( A  e.  V  ->  ( ~P ~P A  =  A  ->  ~P ~P A  ~<  ~P ~P A ) )
101, 9mtoi 178 . 2  |-  ( A  e.  V  ->  -.  ~P ~P A  =  A )
1110neqned 2646 1  |-  ( A  e.  V  ->  ~P ~P A  =/=  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095   ~Pcpw 3997   class class class wbr 4437    ~< csdm 7517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521
This theorem is referenced by: (None)
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