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Theorem dftr4 4538
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 4534 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 4404 . 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 3469   ~Pcpw 4003   U.cuni 4238   Tr wtr 4533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-v 3108  df-in 3476  df-ss 3483  df-pw 4005  df-uni 4239  df-tr 4534
This theorem is referenced by:  tr0  4544  pwtr  4693  r1ordg  8185  r1sssuc  8190  r1val1  8193  ackbij2lem3  8610  tsktrss  9128
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