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Theorem pwne 4358
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4006. (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 4357 . 2  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
2 eqimss 3392 . . 3  |-  ( ~P A  =  A  ->  ~P A  C_  A )
32necon3bi 2639 . 2  |-  ( -. 
~P A  C_  A  ->  ~P A  =/=  A
)
41, 3syl 16 1  |-  ( A  e.  V  ->  ~P A  =/=  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1725    =/= wne 2598    C_ wss 3312   ~Pcpw 3791
This theorem is referenced by:  pnfnemnf  10707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-rab 2706  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793
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