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Theorem dftr4 4267
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 4263 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 4136 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 244 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   Tr wtr 4262
This theorem is referenced by:  tr0  4273  pwtr  4376  r1ordg  7660  r1sssuc  7665  r1val1  7668  ackbij2lem3  8077  tsktrss  8592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-in 3287  df-ss 3294  df-pw 3761  df-uni 3976  df-tr 4263
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