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Theorem dftr4 4535
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 4531 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 4401 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 252 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    C_ wss 3461   ~Pcpw 3997   U.cuni 4234   Tr wtr 4530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-v 3097  df-in 3468  df-ss 3475  df-pw 3999  df-uni 4235  df-tr 4531
This theorem is referenced by:  tr0  4541  pwtr  4690  r1ordg  8199  r1sssuc  8204  r1val1  8207  ackbij2lem3  8624  tsktrss  9142
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