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Theorem pwtr 4417
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 4415 . . 3  |-  U. ~P A  =  A
21sseq1i 3373 . 2  |-  ( U. ~P A  C_  ~P A  <->  A 
C_  ~P A )
3 df-tr 4304 . 2  |-  ( Tr 
~P A  <->  U. ~P A  C_ 
~P A )
4 dftr4 4308 . 2  |-  ( Tr  A  <->  A  C_  ~P A
)
52, 3, 43bitr4ri 271 1  |-  ( Tr  A  <->  Tr  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    C_ wss 3321   ~Pcpw 3800   U.cuni 4016   Tr wtr 4303
This theorem is referenced by:  r1tr  7703  itunitc1  8301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-pw 3802  df-sn 3821  df-pr 3822  df-uni 4017  df-tr 4304
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