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Theorem pwidg 3403
Description: Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
pwidg  |-  ( A  e.  V  ->  A  e.  ~P A )

Proof of Theorem pwidg
StepHypRef Expression
1 ssid 3019 . 2  |-  A  C_  A
2 elpwg 3398 . 2  |-  ( A  e.  V  ->  ( A  e.  ~P A  <->  A 
C_  A ) )
31, 2mpbiri 166 1  |-  ( A  e.  V  ->  A  e.  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434    C_ wss 2974   ~Pcpw 3390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392
This theorem is referenced by:  pwid  3404  axpweq  3953
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