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

Theorem pwtr 4204
Description: A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
Assertion
Ref Expression
pwtr  |-  ( Tr  A  <->  Tr  ~P A
)

Proof of Theorem pwtr
StepHypRef Expression
1 unipw 4202 . . 3  |-  U. ~P A  =  A
21sseq1i 3173 . 2  |-  ( U. ~P A  C_  ~P A  <->  A 
C_  ~P A )
3 df-tr 4088 . 2  |-  ( Tr 
~P A  <->  U. ~P A  C_ 
~P A )
4 dftr4 4092 . 2  |-  ( Tr  A  <->  A  C_  ~P A
)
52, 3, 43bitr4ri 212 1  |-  ( Tr  A  <->  Tr  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    C_ wss 3121   ~Pcpw 3566   U.cuni 3796   Tr wtr 4087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-uni 3797  df-tr 4088
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator