ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwidg Unicode version

Theorem pwidg 3488
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 3081 . 2  |-  A  C_  A
2 elpwg 3482 . 2  |-  ( A  e.  V  ->  ( A  e.  ~P A  <->  A 
C_  A ) )
31, 2mpbiri 167 1  |-  ( A  e.  V  ->  A  e.  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1461    C_ wss 3035   ~Pcpw 3474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041  df-ss 3048  df-pw 3476
This theorem is referenced by:  pwid  3489  axpweq  4053  baspartn  12054  epttop  12096  isopn3  12131
  Copyright terms: Public domain W3C validator