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Theorem dftr4 4015
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 4011 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 3885 . 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 3078   ~Pcpw 3530   U.cuni 3727   Tr wtr 4010
This theorem is referenced by:  tr0  4021  pwtr  4120  r1ordg  7334  r1sssuc  7339  r1val1  7342  ackbij2lem3  7751  tsktrss  8263
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-v 2729  df-in 3085  df-ss 3089  df-pw 3532  df-uni 3728  df-tr 4011
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