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Theorem dftr4 4119
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 4115 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 3988 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 243 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    C_ wss 3153   ~Pcpw 3626   U.cuni 3828   Tr wtr 4114
This theorem is referenced by:  tr0  4125  pwtr  4225  r1ordg  7446  r1sssuc  7451  r1val1  7454  ackbij2lem3  7863  tsktrss  8379
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 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
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 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-v 2791  df-in 3160  df-ss 3167  df-pw 3628  df-uni 3829  df-tr 4115
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