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Theorem 2pwne 7670
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 7651 . . 3  |-  -.  ~P ~P A  ~<  ~P ~P A
2 canth2g 7668 . . . . 5  |-  ( A  e.  V  ->  A  ~<  ~P A )
3 pwexg 4631 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
4 canth2g 7668 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  ~<  ~P ~P A )
53, 4syl 16 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~<  ~P ~P A
)
6 sdomtr 7652 . . . . 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 4450 . . . 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 2670 1  |-  ( A  e.  V  ->  ~P ~P A  =/=  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   ~Pcpw 4010   class class class wbr 4447    ~< csdm 7512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516
This theorem is referenced by: (None)
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