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Theorem 2pwne 7102
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 7083 . . 3  |-  -.  ~P ~P A  ~<  ~P ~P A
2 canth2g 7100 . . . . 5  |-  ( A  e.  V  ->  A  ~<  ~P A )
3 pwexg 4273 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
4 canth2g 7100 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  ~<  ~P ~P A )
53, 4syl 15 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~<  ~P ~P A
)
6 sdomtr 7084 . . . . 5  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<  ~P ~P A )  ->  A  ~<  ~P ~P A )
72, 5, 6syl2anc 642 . . . 4  |-  ( A  e.  V  ->  A  ~<  ~P ~P A )
8 breq1 4105 . . . 4  |-  ( ~P ~P A  =  A  ->  ( ~P ~P A  ~<  ~P ~P A  <->  A 
~<  ~P ~P A ) )
97, 8syl5ibrcom 213 . . 3  |-  ( A  e.  V  ->  ( ~P ~P A  =  A  ->  ~P ~P A  ~<  ~P ~P A ) )
101, 9mtoi 169 . 2  |-  ( A  e.  V  ->  -.  ~P ~P A  =  A )
11 df-ne 2523 . 2  |-  ( ~P ~P A  =/=  A  <->  -. 
~P ~P A  =  A )
1210, 11sylibr 203 1  |-  ( A  e.  V  ->  ~P ~P A  =/=  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864   ~Pcpw 3701   class class class wbr 4102    ~< csdm 6947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951
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