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Theorem dftr4 4120
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 4116 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 3989 . 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 3154   ~Pcpw 3627   U.cuni 3829   Tr wtr 4115
This theorem is referenced by:  tr0  4126  pwtr  4228  r1ordg  7452  r1sssuc  7457  r1val1  7460  ackbij2lem3  7869  tsktrss  8385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-v 2792  df-in 3161  df-ss 3168  df-pw 3629  df-uni 3830  df-tr 4116
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