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Theorem pwne 4157
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3806. (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 4156 . 2  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
2 eqimss 3209 . . 3  |-  ( ~P A  =  A  ->  ~P A  C_  A )
32necon3bi 2397 . 2  |-  ( -. 
~P A  C_  A  ->  ~P A  =/=  A
)
41, 3syl 14 1  |-  ( A  e.  V  ->  ~P A  =/=  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 2148    =/= wne 2347    C_ wss 3129   ~Pcpw 3574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4118
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-rab 2464  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576
This theorem is referenced by:  pnfnemnf  7989
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