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Theorem dftr4 4092
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 4088 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 3961 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 245 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    C_ wss 3127   ~Pcpw 3599   U.cuni 3801   Tr wtr 4087
This theorem is referenced by:  tr0  4098  pwtr  4198  r1ordg  7418  r1sssuc  7423  r1val1  7426  ackbij2lem3  7835  tsktrss  8351
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-v 2765  df-in 3134  df-ss 3141  df-pw 3601  df-uni 3802  df-tr 4088
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