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Theorem 2pwne 7463
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 7444 . . 3  |-  -.  ~P ~P A  ~<  ~P ~P A
2 canth2g 7461 . . . . 5  |-  ( A  e.  V  ->  A  ~<  ~P A )
3 pwexg 4473 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
4 canth2g 7461 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  ~<  ~P ~P A )
53, 4syl 16 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~<  ~P ~P A
)
6 sdomtr 7445 . . . . 5  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<  ~P ~P A )  ->  A  ~<  ~P ~P A )
72, 5, 6syl2anc 656 . . . 4  |-  ( A  e.  V  ->  A  ~<  ~P ~P A )
8 breq1 4292 . . . 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 )
1110neneqad 2679 1  |-  ( A  e.  V  ->  ~P ~P A  =/=  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970   ~Pcpw 3857   class class class wbr 4289    ~< csdm 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309
This theorem is referenced by: (None)
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