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

Theorem pweq 3436
Description: Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
pweq  |-  ( A  =  B  ->  ~P A  =  ~P B
)

Proof of Theorem pweq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sseq2 3049 . . 3  |-  ( A  =  B  ->  (
x  C_  A  <->  x  C_  B
) )
21abbidv 2206 . 2  |-  ( A  =  B  ->  { x  |  x  C_  A }  =  { x  |  x 
C_  B } )
3 df-pw 3435 . 2  |-  ~P A  =  { x  |  x 
C_  A }
4 df-pw 3435 . 2  |-  ~P B  =  { x  |  x 
C_  B }
52, 3, 43eqtr4g 2146 1  |-  ( A  =  B  ->  ~P A  =  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290   {cab 2075    C_ wss 3000   ~Pcpw 3433
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-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3006  df-ss 3013  df-pw 3435
This theorem is referenced by:  pweqi  3437  pweqd  3438  axpweq  4012  pwexg  4021  pwssunim  4120  ordpwsucexmid  4399  istopg  11759  istopon  11773  eltg  11813  tgdom  11833  ntrval  11871
  Copyright terms: Public domain W3C validator