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Theorem pwtr 4226
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 4224 . . 3  |-  U. ~P A  =  A
21sseq1i 3202 . 2  |-  ( U. ~P A  C_  ~P A  <->  A 
C_  ~P A )
3 df-tr 4114 . 2  |-  ( Tr 
~P A  <->  U. ~P A  C_ 
~P A )
4 dftr4 4118 . 2  |-  ( Tr  A  <->  A  C_  ~P A
)
52, 3, 43bitr4ri 269 1  |-  ( Tr  A  <->  Tr  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   Tr wtr 4113
This theorem is referenced by:  r1tr  7448  itunitc1  8046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-tr 4114
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