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Theorem pwne 4309
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3958. (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 4308 . 2  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
2 eqimss 3345 . . 3  |-  ( ~P A  =  A  ->  ~P A  C_  A )
32necon3bi 2593 . 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 1717    =/= wne 2552    C_ wss 3265   ~Pcpw 3744
This theorem is referenced by:  pnfnemnf  10651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-rab 2660  df-v 2903  df-in 3272  df-ss 3279  df-pw 3746
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