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Theorem pwtr 3983
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 3981 . . 3  |-  U. ~P A  =  A
21sseq1i 2997 . 2  |-  ( U. ~P A  C_  ~P A  <->  A 
C_  ~P A )
3 df-tr 3883 . 2  |-  ( Tr 
~P A  <->  U. ~P A  C_ 
~P A )
4 dftr4 3887 . 2  |-  ( Tr  A  <->  A  C_  ~P A
)
52, 3, 43bitr4ri 206 1  |-  ( Tr  A  <->  Tr  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 102    C_ wss 2945   ~Pcpw 3387   U.cuni 3608   Tr wtr 3882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-uni 3609  df-tr 3883
This theorem is referenced by: (None)
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