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Theorem pwne 4177
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3826. (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 4176 . 2  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
2 eqimss 3230 . . 3  |-  ( ~P A  =  A  ->  ~P A  C_  A )
32necon3bi 2487 . 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 1684    =/= wne 2446    C_ wss 3152   ~Pcpw 3625
This theorem is referenced by:  pnfnemnf  10459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627
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