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Theorem pwne 4603
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4234. (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 4602 . 2  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
2 eqimss 3541 . . 3  |-  ( ~P A  =  A  ->  ~P A  C_  A )
32necon3bi 2683 . 2  |-  ( -. 
~P A  C_  A  ->  ~P A  =/=  A
)
41, 3syl 16 1  |-  ( A  e.  V  ->  ~P A  =/=  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1823    =/= wne 2649    C_ wss 3461   ~Pcpw 3999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-rab 2813  df-v 3108  df-in 3468  df-ss 3475  df-pw 4001
This theorem is referenced by:  pnfnemnf  11329
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