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Theorem dftr4 4249
Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
dftr4  |-  ( Tr  A  <->  A  C_  ~P A
)

Proof of Theorem dftr4
StepHypRef Expression
1 df-tr 4245 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 4118 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 244 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    C_ wss 3264   ~Pcpw 3743   U.cuni 3958   Tr wtr 4244
This theorem is referenced by:  tr0  4255  pwtr  4358  r1ordg  7638  r1sssuc  7643  r1val1  7646  ackbij2lem3  8055  tsktrss  8570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-v 2902  df-in 3271  df-ss 3278  df-pw 3745  df-uni 3959  df-tr 4245
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