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Theorem pwne 4193
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3842. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
pwne  |-  ( A  e.  V  ->  ~P A  =/=  A )

Proof of Theorem pwne
StepHypRef Expression
1 pwnss 4192 . 2  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
2 eqimss 3243 . . 3  |-  ( ~P A  =  A  ->  ~P A  C_  A )
32necon3bi 2500 . 2  |-  ( -. 
~P A  C_  A  ->  ~P A  =/=  A
)
41, 3syl 15 1  |-  ( A  e.  V  ->  ~P A  =/=  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1696    =/= wne 2459    C_ wss 3165   ~Pcpw 3638
This theorem is referenced by:  pnfnemnf  10475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640
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