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Theorem 2pwne 7264
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 7245 . . 3  |-  -.  ~P ~P A  ~<  ~P ~P A
2 canth2g 7262 . . . . 5  |-  ( A  e.  V  ->  A  ~<  ~P A )
3 pwexg 4384 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
4 canth2g 7262 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  ~<  ~P ~P A )
53, 4syl 16 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~<  ~P ~P A
)
6 sdomtr 7246 . . . . 5  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<  ~P ~P A )  ->  A  ~<  ~P ~P A )
72, 5, 6syl2anc 644 . . . 4  |-  ( A  e.  V  ->  A  ~<  ~P ~P A )
8 breq1 4216 . . . 4  |-  ( ~P ~P A  =  A  ->  ( ~P ~P A  ~<  ~P ~P A  <->  A 
~<  ~P ~P A ) )
97, 8syl5ibrcom 215 . . 3  |-  ( A  e.  V  ->  ( ~P ~P A  =  A  ->  ~P ~P A  ~<  ~P ~P A ) )
101, 9mtoi 172 . 2  |-  ( A  e.  V  ->  -.  ~P ~P A  =  A )
1110neneqad 2675 1  |-  ( A  e.  V  ->  ~P ~P A  =/=  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    =/= wne 2600   _Vcvv 2957   ~Pcpw 3800   class class class wbr 4213    ~< csdm 7109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113
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