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Theorem pwtr 4656
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 4653 . . 3  |-  U. ~P A  =  A
21sseq1i 3491 . 2  |-  ( U. ~P A  C_  ~P A  <->  A 
C_  ~P A )
3 df-tr 4497 . 2  |-  ( Tr 
~P A  <->  U. ~P A  C_ 
~P A )
4 dftr4 4501 . 2  |-  ( Tr  A  <->  A  C_  ~P A
)
52, 3, 43bitr4ri 278 1  |-  ( Tr  A  <->  Tr  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    C_ wss 3439   ~Pcpw 3971   U.cuni 4202   Tr wtr 4496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-pw 3973  df-sn 3989  df-pr 3991  df-uni 4203  df-tr 4497
This theorem is referenced by:  r1tr  8098  itunitc1  8704
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